3 Monotonicity

Monotonicity at a Point

(a) A function f(x) is called an increasing function at point x = a, if in a sufficiently small

neighbourhood of x = a ; f(a h) < f(a) < f(a + h)

(b) A function f(x) is called a decreasing function at point x = a, if in a sufficiently small

neighbourhood of x = a ; f(a h) > f(a) > f(a + h)

Note: If x = a is a boundary point, then use the appropriate one sides inequality to test monotonicity
of f(x).

(c) Testing of Monotonicity of Differentiable Function at A Point.

(i) If f '(a) > 0, then f(x) is increasing at x = a.
(ii) If f '(a) < 0, then f(x) is decreasing at x = a.
(iii) If f '(a) = 0, then examine the sign of f '(a+) and f '(a ).
(1) If f '(a+) > 0 and f '(a ) > 0, then increasing
(2) If f '(a+) < 0 and f '(a ) < 0, then decreasing
(3) otherwise neither increasing nor decreasing.

Monotonocity 69
 Example 1:
Let f(x) = x3 3x + 2. Examine the nature of function at points x = 0, 1 & 2.
Solution:

f(x) = x3 3x + 2

f '(x) = 3(x2 1) = 0
 x = ±1
(i) f '(0) = 3  decreasing at x = 0
(ii) f '(1) = 0
also, f '(1+) = positive and f'(1 ) = negative
 neither increasing nor decreasing at x = 0.
(iii) f '(2) = 9  increasing at x = 2

Concept Builders - 1

(i) If function f(x) = x3 + x2 x + 1 is increasing at x = 0 & decreasing at x = 1, then find the greatest
integral value of .

Monotonicity Over an Interval

(a) A function f(x) is said to be monotonically increasing (MI) in (a, b) if f '(x)  0 where equality
holds only for discrete values of x i.e. f '(x) does not identically become zero for x  (a, b) or
any sub interval.
(b) f(x) is said to be monotonically decreasing (MD) in (a, b) if f'(x) 0 where equality holds only for
discrete values of x i.e. f'(x) does not identically become zero for x  (a, b) or any sub interval.

Note: (i) A function is said to be monotonic if it's either increasing or decreasing.

(ii) If a function is invertible it has to be either increasing or decreasing.

Example 2:

( )
Prove that the function f(x) = log x3 + x6 + 1 is entirely increasing.

Solution:

(
Now, f(x) = log x3 + x6 + 1 )
1  2 x5  3x2
f '(x) =  3x +  = >0
x 3 + x6 + 1  2 x6 + 1  x6 + 1
 f(x) is increasing.
Example 3:
Find the intervals of monotonicity of the function y = x2 loge|x|, (x  0).

Solution:

Let y = f(x) = x2 loee|x|

70 Monotonocity
 1
f '(x) = 2x ; for all x (x  0)
x

f '(x) =
2x2 − 1
 f '(x) =
( )(
2x − 1 )
2x + 1
x x
 1   1 
So f ' (x) > 0 when x   − ,0    ,   and f '(x) < 0
 2   2 
 1   1 
when x   −, −    0, 
 2  2
 1   1 
f(x) is increasing when x   − ,0    ,
 2   2 
 1   1 
and decreasing when x   −, −    0,  Ans.
 2  2

Concept Builders - 2

  
(i) If f(x) = sinx + n |secx + tanx| 2x for x   − ,  then check the monotonocity of f(x)
 2 2
(ii) Prove that y = ex + sinx is increasing in x  R+

Greatest and Least Value of a Function

(a) Extreme Value Theorem: If f is continuous on [a, b] then f takes on, a least value m and
a greatest value M on this interval.

NOTE:
(a) Continuity through the interval [a,b] is essential for the validity of this theorem. There is
a discontinuity at x = c, c  [a,b]. The function has a minimum value at the x = a and has
no maximum value.

Monotonocity 71
 (b) If a continuous function y = f(x) is increasing in the closed interval [a, b], then f(a) is the
least value and f(b) is the greatest value of f(x) in [a, b] (figure-1)

(c) If a continuous function y = f(x) is decreasing in [a, b], then f(b) is the least and f(a) is
the greatest value of f(x) in [a, b]. (figure-2)

(d) If a continuous function y = f(x) is increasing/decreasing in the (a, b), then no greatest
and least value exist.

Example 4:
x  1 
Show that f(x) = sin 1
nx is decreasing in x   , 3  . Also find its range.
1 + x2  3 
Solution:

f(x) = sin 1 x
nx = tan x 1
nx  f '(x) =
1 1
=
(
− 1 + x2 − x )
1+ x 2 1 + x2 x x 1 + x2( )
 1 
 f '(x) < 0  x   , 3
 3 
 f(x) is decreasing.
 1 
f(x)|max = f 
 3
 =

6
1
+ n3 and f(x)|min = f
2
( 3 ) = 3 1
2
n3

 1  1 
 Range of f(x) =  − n3, + n3 Ans.
3 2 6 2 

Example 5:
Find the greatest and least value of f(x) = x3 + 5x + ex in [1, 3]
Solution:
f '(x) = 3x2 + 5 + ex f(x) is always increasing.

72 Monotonocity
 Least value = f(1) = 6 + e
greatest value = f(3) = (42 + e3)

Concept Builders - 3

x 3 x2
(i) Let f(x) = − + 2 in [ 2, 2]. Find the greatest and least value of f(x) in [ 2, 2]
3 2

Proving Inequalities Using Monotonicity

Comparison of two functions f(x) and g(x) can be done by analyzing their monotonic behavior.

Significance of the Sign of IInd Order Derivative

The sign of the 2nd order derivative determines the concavity of the curve.
If f ''(x) > 0  x  (a, b) then graph of f(x) is concave upward in (a, b).
Similarly, if f ''(x) < 0  x  (a, b) then graph of f(x) is concave downward in (a, b).

Rolle's Theorem
Let f be a function that satisfies the following three conditions:
(a) f is continuous on the closed interval [a, b].
(b) f is differentiable on the open interval (a, b)
(c) f(a) = f(b)
Then there exist at least one number c in (a, b) such that f '(c) = 0.

Note: If f is differentiable function then between any two consecutive roots of f(x) = 0, there is at
least one root of the equation f '(x) = 0.
(d) Geometrical Interpretation:
Geometrically, the Rolle's theorem says that somewhere between A and B the curve has at
least one tangent parallel to x-axis.

Monotonocity 73
Example 6:
Verify Rolle's theorem for the function f(x) = x3 3x2 + 2x in the interval [0, 2].
Solution:
Here we observe that
(a) f(x) is polynomial and since polynomial are always continuous, as well as differentiable.
Hence f(x) is continuous in the [0,2] and differentiable in the (0, 2).
and
(b) f(0) = 0, f(2) = 23
3. (2) + 2(2) = 0
2

 f(0) = f(2)
Thus, all the condition of Rolle's theorem are satisfied.
So, there must exists some c  (0, 2) such that f'(c) = 0
1
 f '(c) = 3c2 6c + 2 = 0  c = 1 ± 1
3
1
where both c = 1 ±  (0, 2) thus Rolle's theorem is verified.
3

Example 7:
4
Let Rolle's theorem holds for f(x) = x3 + bx2 + ax, when 1  x  2 at the point c = , then find
3
a + b.
Solution:
f(1) = f(2) 1 + b + a = 8 + 4b + 2a
a + 3b + 7 = 0 .......(1)
f '(c) = 3x2 + 2bx + a = 0
16 8b
+ + a = 0  3a + 8b + 16 = 0 .........(2)
3 3
By solving a = 8, b = 5

Concept Builders - 4

(i) Verify Rolle's theorem for y = 1 x4/3 on the interval [ 1,1]

(ii) (a) Let f(x) = 1 x2/3. Show that f( 1) = f(1) but there is no number c in ( 1, 1) such that
f '(c) = 0.
Why does this not contradict Rolle's Theorem ?
(b) Let f(x) = (x 1) 2. Show that f(0) = f(2) but there is no number c in (0, 2) such that
f '(c) = 0.
Why does this not contradict Rolle's Theorem ?

74 Monotonocity
Lagrange's Mean Value Theorem (LMVT)

Let f be a function that satisfies the following conditions:

(i) f is continuous in [a, b]
(ii) f is differentiable in (a, b).
f(b) − f(a)
Then there is a number c in (a, b) such that f '(c) =
b−a
(a) Geometrical Interpretation:
Geometrically, the Mean Value Theorem says that somewhere between A and B the curve
has at least one tangent parallel to chord AB.
(b) Physical Interpretations:
If we think of the number (f(b) f(a))/(b a) as the average change in f over [a, b] and
f'(c) as an instantaneous change, then the Mean Value Theorem says that at some
interior point the instantaneous change must equal the average change over the entire
interval.

Example 8:
Find c of the Lagrange's mean value theorem for the function f(x) = 3x2 + 5x + 7 in the interval
[1, 3].
Solution:
Given f(x) = 3x2 + 5x + 7 ...... (i)
 f(1) = 3 + 5 + 7 = 15 and f(3) = 27 + 15 + 7 = 49
Again f '(x) = 6x + 5
Here a = 1, b = 3
Now from Lagrange's mean value theorem
f(b) − f(a) f(3) − f(1) 49 − 15
f '(c) =  6c + 5 = = = 17 or c = 2.
b−a 3−1 2

Example 9:
If f(x) is continuous and differentiable over [ 2, 5] and 4  f '(x)  3 for all x in
( 2, 5), then the greatest possible value of f(5) f( 2) is -
(A) 7 (B) 9 (C) 15 (D) 21

Monotonocity 75
Solution:
Apply LMVT
f(5) − f(−2)
f '(x) = for some x in ( 2, 5)
5 − (−2)
f(5) − f(−2)
Now, 4 3
7
28  f(5) f( 2)  
 Greatest possible value of f(5) f( 2) is 21.

Concept Builders - 5

(i) If f(x) = x2 in [a, b], then show that there exist atleast one c in (a, b) such that a, c, b are in A.P.
(ii) Find C of LMVT for f(x) = |x|3 in [2, 5].

Special Note
Use of Monotonicity in identifying the number of roots of the equation in a given interval. Suppose a
and b are two real numbers such that,
(a) Let f(x) is differentiable & either MI or MD for 0  x  b.
and
(b) f(a) and f(b) have opposite signs.
Then there is one & only one root of the equation f(x) = 0 in (a, b).

Miscellaneous Examples

Example 10:
If g(x) = f(x) + f(1 x) and f ''(x) < 0; 0  x  1, show that g(x) increasing in x  (0, 1/2) and
decreasing in x  (1/2, 1).
Solution:
 f ''(x) < 0  f '(x) is decreasing function.
Now, g(x) = f(x) + f(1 x)
 g'(x) = f '(x) f '(1 x) ......... (i)
Case I:
If x > (1 x)
 x > 1/2
 f '(x) < f '(1 x)
 f '(x) f '(1 x) < 0
 g'(x) < 0
1 
 g(x) decreases in x   , 1
2 

76 Monotonocity
 Case II:
If x < (1 x) x < 1/2
 f '(x) > f '(1 x)
 f '(x) f '(1 x) > 0
 g'(x) > 0
 g(x) increases in x  (0, 1/2)

Example 11:
 
Which of the following functions are decreasing on  0, 
 2
(A) cos x (B) cos2x (C) cos3x (D) tan x
Solution:

f(x) = cosx

f(x) = cos2x

f(x) = cos3x Non-monotonic

 
f(x) = tanx is increasing in  0,  Option A and B are correct.
 2

Example 12:
Prove that the equation e(x 1)
+ x = 2 has one solution
Solution:
Let f(x) = e(x 1)
+x
f '(x) = e (x 1)
+1
f(x) is always an increasing function
lim f(x) =  and lim f(x) = 0
x → x →

f(x) = 2 has exactly one solution.

Monotonocity 77
 ANSWER KEY FOR CONCEPT BUILDER

1. (i) 4

2. (i) Increasing

8 8
3. (i) Greatest is and least value is .
3 3

4. (i) Rolle's theorem is valid

(ii) (a) f(x) is non-differentiable at x = 0 in ( 1,1)

(b) f(x) is discontinuous at x = 1 in (0,2)

117
5. (ii) C=
9

78 Monotonocity
 Objective Exercise - I

1. If the function f (x) = 2 x2 kx + 5 is increasing in [1, 2], then ' k ' lies in the interval
(A) ( , 4) (B) (4, ) (C) ( , 8] (D) (8, )

2. f(x) = x + 1/x, x  0 is monotonic increasing when-

(A) |x| < 1 (B) |x| > 1 (C) |x| < 2 (D) |x| > 2

3. The function xx decreases on the interval-

 1
(A) (0, e) (B) (0, 1) (C)  0,  (D) None of these
 e
4. Function f(x) = x2(x 2)2 is-
(A) increasing in (0, 1)  (2, ) (B) decreasing in (0, 1)  (2, )
(C) decreasing function (D) increasing function

5. If f and g are two decreasing function such that fog is defined, then fog will be-
(A) increasing function (B) decreasing function
(C) neither increasing nor decreasing (D) None of these

6. If function f(x) = 2x2 + 3x m log x is monotonic decreasing in the interval (0, 1), then the least
value of the parameter m is-
15 31
(A) 7 (B) (C) (D) 8
2 4
x 2
7. If f(x) = + for 7  x  7, then f(x) is monotonic increasing function of x in the interval-
2 x
(A) [7, 0] (B) [2, 7] (C) [ 2, 2] (D) [0, 7]

8. If f(x) = x3 10x2 + 200x 10, then f(x) is-

(A) decreasing in ( , 10] and increasing in (10,)
(B) increasing in ( , 10] and decreasing in (10,)
(C) increasing for every value of x
(D) decreasing for every value of x

9. The value of K in order that f(x) = sinx cosx Kx + b decreases for all real values is given by-
(A) K < 1 (B) K 1 (C) K  2 (D) K < 2

10. When 0  x  1, f(x) = |x| + |x 1| is-

(A) increasing (B) decreasing (C) constant (D) None of these

11. If 2a + 3b + 6c = 0, then at least one root of the equation ax 2+ bx + c = 0 lies in the interval-
(A) (0, 1) (B) (1, 2) (C) (2, 3) (D) none

Monotonocity 79
12. The greatest value of x3 18x2+ 96x in the interval (0, 9) is-
(A) 128 (B) 60
(C) 160 (D) 120

13. Difference between the greatest and the least values of the function f(x) = x(n x 2) on [1, e2]
is
(A) 2 (B) e
(C) e 2
(D) 1

14. A maximum point of 3x4 2x3 6x2+ 6x + 1 in [0, 2] is-

(A) x = 0 (B) x = 1
(C) x = 1/2 (D) does not exist

nx
15. Range of the function f(x) = is
x
(A) ( , e) (B) ( , e2)
 2  1
(C)  −,  (D)  −, 
 e  e


16. f(x) = 1 + [cosx]x, in 0  x 
2
(where [.] denotes greatest integer function)
(A) has a minimum value 0 (B) has a maximum value 2
  
(C) is continuous in 0,  (D) is not differentiable at x =
 2 2

17. A value of C for which the conclusion of Mean values theorem holds for the function
f(x) = logex on the interval [1, 3] is-
1
(A) 2log3e (B) loge3
2
(C) log3e (D) loge3

  1
x cos   , x  0
18. The value of c in Lagrange's theorem for the function f(x) =  x in the interval
 0, x = 0

[ 1, 1] is:
1
(A) 0 (B)
2
1
(C) (D) Non-existent in the interval
2

80 Monotonocity
19. If the function f(x) = x3 6x2+ ax + b defined on [1, 3], satisfies the rolle's theorem
2 3+1
for c = then-
3
(A) a = 11, b = 6 (B) a = 11, b = 6
(C) a = 11, b  R (D) None of these

20. The function f: [a, ) →R where R denotes the range corresponding to the given domain, with
rule f(x) = 2x3 3x2+ 6, will have an inverse provided
(A) a 1 (B) a 0
(C) a 0 (D) a 1

21. If the function f (x) = 2x2+ 3x + 5 satisfies LMVT at x = 2 on the closed interval [1, a], then the
value of 'a' is equal to:
(A) 3 (B) 4
(C) 6 (D) 1

ANSWER KEY

1. (A) 2. (B) 3. (C) 4. (A) 5. (A) 6. (A) 7. (B)

8. (C) 9. (C) 10. (C) 11. (A) 12. (C) 13. (B) 14. (C)

15. (C) 16. (C) 17. (A) 18. (D) 19. (C) 20. (A) 21. (A)

Monotonocity 81
 Objective Exercise - II

Single Correct Type Questions

1. Let f (x) and g (x) be two continuous functions defined from R → R, such that f(x1) > f(x2) and g
(x1) < g (x2),  x1 > x2, then solution set of f(g(2 2)) > f(g(3 4)) is:
(A) R (B)  (C) (1, 4) (D) R [1, 4]

2. If the equation anxn + an 1xn 1 + .... + a1x = 0 has a positive root x = , then the equation
nanxn 1 + (n 1) an 1xn 2 + .... + a1= 0 has a positive root, which is:
(A) Smaller than  (B) Greater than 
(C) Equal to  (D) Greater than or equal to 

 1 − x2 
3. The function f (x) = tan 1  2 
is -
1+ x 
(A) increasing in its domain
(B) decreasing in its domain
(C) decreasing in ( , 0) and increasing in (0,)
(D) increasing in ( , 0) and decreasing in (0,)

d
4. Given f '(1) = 1 and (f (2x)) = f '(x)  x > 0. If f '(x) is differentiable then there exists a number
dx
c  (2, 4) such that f '' (c) equals
(A) 1/4 (B) 1/8 (C) 1/4 (D) 1/8

5. Statement-1: The function x2 (ex + e x) is increasing for all x > 0.

Statement-2: The functions x2ex and x2e x are increasing for all x > 0 and the sum of two
increasing functions in any interval (a, b) is an increasing function in (a, b).
(A) Statement-1 is false; Statement-2 is true.
(B) Statement-1 is true; Statement-2 is false.
(C) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for
Statement-1
(D) Statement-1 is true; Statement-2 is true; Statement 2 is not a correct explanation for
Statement-1

6. Let f (x) and g (x) are two function which are defined and differentiable for all x x0.
If f(x0) = g (x0) and f ' (x) > g ' (x) for all x > x0 then
(A) f (x) < g (x) for some x > x0 (B) f (x) = g (x) for some x > x0
(C) f (x) > g (x) only for some x > x0 (D) f (x) > g (x) for all x > x0

One or More Than One Correct Type Questions

7. Let y = f(x) be a bijective function and differentiable  x  R then which of the following is/are
correct?
(A) y = f 1(x) is differentiable if y = f(x) is always concave up
(B) y = f 1(x) is differentiable if y = f(x) is always concave down
(C) y = f 1(x) is differentiable if f '(x) = 0 has no real roots
(D) None of these

82 Monotonocity
8. Let f(x) is a derivable function, which is increasing for all x R (having no critical point), then:
(A) f(3 4x) is an increasing function for all x.
(B) f(3 4x) is a decreasing function for all x.
1
(C) f(x2 x) increasing for x >
2
(D) (f(x))3is an increasing function for all x.

9. Let g'(x) > 0 and f '(x) < 0,  x  R, then

(A) g (f(x +1)) > g (f(x 1)) (B) f(g(x 1)) > f(g (x + 1))
(C) g(f (x +1)) < g(f(x 1)) (D) g(g(x + 1)) < g(g(x 1))

10. If f(x) = x3 x2 + 100x + 1001, then

 1   1 
(A) f(2000) > f(2001) (B) f  > f 
 1999   2000 
(C) f(x + 1) > f(x 1) (D) f (3x 5) > f(3x)

11. If the derivative of an odd cubic polynomial vanishes at two different value of 'x' then
(A) coefficient of x3and x in the polynomial must be same in sign
(B) coefficient of x3and x in the polynomial must be different in sign
(C) the values of 'x' where derivative vanishes are closer to origin as compared to the respective
roots on either side of origin.
(D) the values of 'x' where derivative vanishes are far from origin as compared to the respective
roots on either side of origin.

12. Let h(x) = f(x) (f(x))2 + (f(x))3for every real number 'x' and f(x) is a differentiable function,
then
(A) 'h' is increasing whenever 'f' is increasing
(B) 'h' is increasing whenever 'f' is decreasing
(C) 'h' is decreasing whenever 'f' is decreasing
(D) Nothing can be said in general

13. The set of all x for which the function h(x) = log2( 2x 3 + x2) is defined and monotonic, is
(A) (1, 3) (B) ( , 1) (C) ( 1, 1) (D) (3,)

3x2 + 12x − 1, −1  x  2
14. If f (x) =  then:
 37 − x, 2x3

(A) f(x) is increasing on [ 1, 2] (B) f(x) is continuous on [ 1, 3]

(C) f (D) f(x) has the maximum value at x = 2

Monotonocity 83
15. Which of the following is/are correct?
(A) Between any two roots of excosx = 1, there exists at least one root of tanx = 1.
(B) Between any two roots of exsinx = 1, there exists at least one root of tanx = 1.
(C) Between any two roots of e cosx = 1, there exists at least one root of e sinx = 1.
x x

(D) Between any two roots of exsinx = 1, there exists at least one root of excosx = 1.

1 
2/3  tan[x]

16. Given: f (x) = 4  − x g(x) =  x , x  0 h(x) = {x} k(x) = 5log2 (x + 3) then in [0, 1], Lagrange's
2   1, x=0
Mean Value Theorem is NOT applicable to
(A) f, g, h (B) h, k (C) f, g (D) g, h, k
where [x] and {x} denotes the greatest integer and fractional part function.

Comprehension # 1
 x + sin x
Consider a function f defined by f(x) = sin 1 sin   ,  x  [0, ], which satisfies
 2 
f(x) + f(2 x) = ,  x  [, 2] and f(x) = f(4 x) for all x  [2, 4], then

17. If  is the length of the largest interval on which f(x) is increasing, then  =

(A) (B)  (C) 2 (D) 4
2

18. If f(x) is symmetric about x = , then  =

 
(A) (B)  (C) (D) 2
2 4

19. Maximum value of f(x) on [0, 4] is:

 
(A) (B)  (C) (D) 2
2 4

ANSWER KEY

1. (C) 2. (A) 3. (D) 4. (B) 5. (B) 6. (D) 7. (ABC)

8. (BCD) 9. (BC) 10. (BC) 11. (BC) 12. (AC) 13. (BD) 14. (ABCD)

15. (ABCD) 16. (A) 17. (C) 18. (B) 19. (A)

84 Monotonocity
 Subjective Exercise - I

1. Find the intervals of monotonocity for the following functions & represent your solution set on

the number line.

2
(A) f(x) = 2. ex − 4x
(B) f(x) = ex/x (C) f(x) = x2e x
(D) f(x) = 2x2 n | x |

2. Find the intervals of monotonocity of the functions in [0, 2]

(A) f (x) = sin x cos x in x [0, 2] (B) g (x) = 2 sinx + cos 2x in (0  x  2).

3. Find the greatest & the least values of the following functions in the given interval if they exist.

x  1 
(A) f(x) = sin 1
n x in  , 3
x2 + 1  3 

(B) f(x) = 12x4/3 6x1/3, x  [ 1, 1]

(C) y = x5 5x4 + 5x3 + 1 in [ 1, 2]

4. Find the set of values of x for which the inequality n (1 + x) > x/(1 + x) is valid.

5. Verify Rolle's theorem for f(x) = (x a)m(x b)n on [a, b] ; m, n being positive integer.

6. Let f(x) = 4x3 3x2 2x + 1, use Rolle's theorem to prove that there exist a "c", 0 < c <1 such

that f(c) = 0.

7. f(x) and g(x) are differentiable functions for 0  x  2 such that f(0) = 5, g(0) = 0, f(2) = 8, g(2) = 1.

Show that there exists a number c satisfying 0 < c < 2 and f ' (c) = 3 g' (c).

 3 x=0
8. For what value of a, m and b does the function f (x) =  − x2
+ 3x + a 0 x1

mx + b 1 x 2

satisfy the hypothesis of the mean value theorem for the interval [0, 2].

9. Let f (x) be a increasing function defined on (0, ). If f(2a2+ a + 1) > f(3a2 4a + 1). Find the range

of a.

Monotonocity 85
 ANSWER KEY

1. (A) I in (2, ) and D in ( , 2) (B) I in (1, ) and D in ( , 0)  (0, 1)

1 1 1 1
(C) I in (0, 2) and D in ( , 0)  (2,) (D) I for x > or < x < 0 and D for x < or 0 < x <
2 2 2 2

2. (A) I in [0, 3/4)  (7/4, 2] and D in (3/4, 7 /4)

(B) I in [0, /6)  (/2, 5/6) (3/2 , 2] and D in (/6, /2)  (5/6, 3 /2)]

(C) I in [0, /2)  (3/2, 2] and D in (/2, 3/2)

3. (A) (/6) + (1/2)ln 3, (/3) (1/2)ln 3

(B) 9/4

(C) 2 and 10

mb + na
4. ( 1, 0)  (0,) 5. c = which lies between a and b
m+n

8. a = 3, b = 4 and m = 1 9. (0, 1/3) (1, 5)

86 Monotonocity
 Subjective Exercise - II

1. Let f(x) = 1 x x3. Find all real values of x satisfying the inequality, 1 f(x) f3(x) > f(1 5x)

max f(t) : 0  t  x , 0  x  1
2. Let f (x) = x3 x2 + x + 1 and g(x) = 
 3 − x, 1 x 2

Discuss the continuity & differentiability of g(x) in the interval (0, 2).

3. If f (x) = 2ex ae x
+ (2a + 1)x 3 monotonically increases for every x R then find the range

4. (a) Let f, g be differentiable on R and suppose that f(0) = g(0) and f '(x)  g'(x) for all x  0. Show
that f(x)  g(x) for all x 0.
(b) Show that exactly two real values of x satisfy the equation x2 = xsinx + cosx.
(c) Prove that inequality ex > (1 + x) for all x  R0 and use it to determine which of the two
numbers e and e is greater.

5. Assume that f is continuous on [a, b], a > 0 and differentiable on an open interval
f(a) f(b)
(a, b). Show that if = , then there exist x0  (a, b) such that x0f '(x0) = f(x0).
a b

6. Find all the values of the parameter 'a' for which the function ;
f(x) = 8ax a sin 6x 7x sin 5x increasing and has no critical points for all x R.

1
7. Find the values of 'a' for which the function f(x) = sin x a sin2x sin3x + 2ax increases
3
throughout the number line.

 1
8. Find the minimum value of the function f(x) = x3/2 + x 3/2
4  x +  for all permissible real x.
 x

9. Find the set of values of 'a' for which the function,

 21 − 4a − a2 
f(x) =  1 −  x3 + 5x + 7 is increasing at every point of its doMain.
 a+1 
 

10. If f, ,  are continuous in [a, b] and derivable in ]a, b[ then show that there is a value of c lying
between a & b such that,
f(a) f(b) f '(c)
(a) (b)  '(c) = 0
(a) (b)  '(c)

Monotonocity 87
11. Prove that, x2 1 > 2x n x > 4(x 1) 2 n x for x > 1.

12. Prove that if f is differentiable on [a, b] and if f (a) = f (b) = 0 then for any real  there is an
x (a, b) such that f (x) + f ' (x) = 0.

13. Let a > 0 and f be continuous in [ a, a]. Suppose that f '(x) exists and f '(x)  1 for all
x ( a, a). If f(a) = a and f( a) = a, show that f(0) = 0.

14. Let f be continuous on [a, b] and differentiable on (a, b). If f (a) = a and f (b) = b, show that
there exist distinct c1, c2 in (a, b) such that f ' (c1) + f '(c2) = 2.

ANSWER KEY

1. ( 2, 0)  (2,) 2. continuous but not diff. at x = 1 3. a 0

6. (6,) 7. [1,) 8. 10

9. [ 7, 1) [2, 3]

88 Monotonocity
 JEE(Main)-(Previous Year Questions)

1. A function is matched below against an interval where it is supposed to be increasing. which of

the following pairs is incorrectly matched? [AIEEE-2005]
interval function
(1) ( ,) x3 3x2 + 3x + 3
(2) [2, ) 2x3 3x2 12x + 6
 1
(3)  −,  3x2 2x + 1
 3
(4) ( , 4) x3 + 6x2+ 6

2. The function f(x) = tan 1 (sinx + cosx) is an increasing function in- [AIEEE-2007]
(1) (/4, /2) (2) ( /2, /4) (3) (0, /2) (4) ( /2, /2)

3. If f and g are differentiable functions in [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 and f(1) = 6, then
for some c  ] 0, 1 [: [JEE(Main)-2014]
(1) 2f '(c) = g'(c) (2) 2f '(c) = 3g'(c)
(3) f '(c) = g'(c) (4) f '(c) = 2g '(c)

x d−x
4. Let f(x) = − , x  R , where a, b and d are non-zero real constants. Then:
a 2 + x2 b2 + (d− x)2

(1) f is an increasing function of x [JEE(Main)-2019]

(2) f is a decreasing function of x
(3) f is neither increasing nor decreasing function of x
(4) f' is not a continuous function of x

5. If the function f given by f(x) = x3 3(a 2)x2 + 3ax + 7, for some a  R is increasing in (0, 1] and
−
decreasing in [1, 5), then a root of the equation, = 0 (x  1) is: [JEE(Main)-2019]
(x− 1)2
(1) 6 (2) 7 (3) 5 (4) 7

6. Let f: [0, 2] → R be a twice differentiable function such that f "(x) > 0, for all x  (0, 2).
If (x) = f(x) + f(2 x), then  is [JEE(Main)-2019]
(1) decreasing on (0, 2)
(2) increasing on (0, 2)
(3) increasing on (0, 1) and decreasing on (1, 2)
(4) decreasing on (0, 1) and increasing on (1, 2)

Monotonocity 89
7. Let f(x) = ex x and g(x) = x2 x,  x  R. Then the set of all x  R, where the function
h(x) = (f o g) (x) is increasing is: [JEE(Main)-2019]
 1    
(1) 0,   [1, ) (2)     
 2  2  2 
 
(3) [0, ) (4)  ,0  1,  )
2 

8. Let the function, f: [ 7, 0] → R be continuous on [ 7, 0] and differentiable on ( 7, 0). If

f( 7) = 3 and f'(x)  2, for all x  ( 7, 0), then for all such functions f, f( 1) + f(0) lies in the
interval: [JEE(Main)-2020]
(1) [ 3, 11] (2) ( , 20] (3) ( , 11] (4) [ 6, 20]

9. The value of c in the Lagrange's mean value theorem for the function f(x) = x3 4x2 + 8x + 11,
when x  [0,1] is: [JEE(Main)-2020]
2 7 −2 4− 5 4− 7
(1) (2) (3) (4)
3 3 3 3
  
10. Let f(x) = x cos 1 ( sin|x|), x   − ,  then which of the following is true?
 2 2

(1) =−
2
    
(2) f' is decreasing in  − ,0  and increasing in  0, 
 2   2
(3) f is not differentiable at x = 0
    
(4) f' is increasing in  − ,0  and decreasing in  0,  [JEE(Main)-2020]
 2   2

 x2 +  
11. If c is a point at which Rolle's theorem holds for the function, =   in the interval
 7x 
e

[3, 4], where a  R, then f"(c) is equal to: [JEE(Main)-2020]

1 1 3 1
(1) (2) − (3) (4) −
12 12 7 24

12. Let f be any function continuous on [a, b] and twice differentiable on (a, b). If for all x  (a, b),
−
f'(x) > 0 and f" (x) < 0, then for any c  (a, b), is greater than: [JEE(Main)-2020]
−
b−c a +b c−a
(1) (2) 1 (3) (4)
c−a b−a b−c

13. Let f be a twice differentiable function on (1, 6). If f(2) = 8, f'(2) = 5, f'(x)  1 and f"(x)  4, for all
x  (l, 6), then: [JEE(Main)-2020]
(1) f'(5) + f"(5)  20 (2) f(5) 10 (3) f(5) + f'(5)  28 (4) f(5) + f'(5)  26

90 Monotonocity
 4
14. 3
ax2 + bx 4, x    = 0, then
3
ordered pair (a, b) is equal to : [JEE(Main)-2021]
(1) ( 5, 8) (2) (5, 8) (3) (5, 8) (4) ( 5, 8)

15. Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10
lie in the interval (a, a + 1) Then, |a| is equal to
[JEE(Main)-2021]

16. The number of real solutions of x7 + 5x3 + 3x + 1 = 0 is equal to ______.

[JEE(Main)-2021]
(1) 0 (2) 1 (3) 3 (4) 5

 1  1
17. Let f and g be twice differentiable even functions on ( 2, 2) such that f   = 0, f   = 0 ,
4 2
3
f(1) = 1 and g   = 0 , g(1) = 2. Then, the minimum number of solutions of f(x) g'' 'x = 0
4
in ( 2, 2) is equal to ____. [JEE(Main)-2021]


 x − x + 10x − 7, x  1
3 2

18. Let f(x) = 


 ( )
−2x + log 2 + b2 − 4 , x  1

Then the set of all values of b, for which f(x) has maximum value at x = 1, is
[JEE(Main)-2021]
(1) ( 6, 2) (2) (2, 6)

(3) [( 6, 2)  (2, 6)] (4) [( 6, 2)  (2, 6 )]

19. The number of distinct real roots of the equation x7 7x 2=0

[JEE(Main)-2022]
(1) 5 (2) 7 (3) 1 (4) 3

20. f(x)= 4loge(x 1) 2x2 + 4x + 5, x > 1, which one of the following is NOT correct?
[JEE(Main)-2022]
(1) f is increasing in (1, 2) and decreasing in (2, )
(2) f(x) = 1 has exactly two solutions

(4) f(x) = 0 has a root in the interval (e, e + 1)

Monotonocity 91
 ANSWER KEY

1. (3) 2. (2) 3. (4) 4. (1) 5. (2) 6. (4) 7. (1)

8. (2) 9. (4) 10. (2) 11. (1) 12. (4) 13. (3) 14. (2)

15. 2 16. (2) 17. 4 18. (3) 19. (4) 20. (3)

92 Monotonocity
 JEE(Advanced)-(Previous Year Questions)

1. If f(x) is a twice differentiable function and given that f(1) = 1, f(2) = 4, f(3) = 9, then
(A) f '' (x) = 2, for x (1, 3) (B) f '' (x) = f ' (x) = 2, for some x (2, 3)
(C) f '' (x) = 3, for x (2, 3) (D) f '' (x) = 2, for some x (1, 3)
[JEE 2005 (Scr)]
2. Let f (x) = 2 + cos x for all real x.
Statement-1: For each real t, there exists a point 'c' in [t, t + ] such that f ' (c) = 0.
because
Statement-2: f(t) = f(t + 2) for each real t.
(A) Statement-1 is true, statement-2 is true; statement-2 is correct explanation for
statement-1.
(B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for
statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true. [IIT JEE 2007]

3. Paragraph [IIT JEE 2007]

If a continuous function f defined on the real line R, assumes positive and negative values in R
then the equation f(x) = 0 has a root in R. For example, if it is known that a continuous function
f on R is positive at some point and its minimum value is negative then the equation
f(x) = 0 has a root in R.
Consider f(x) = kex x for all real x where k is a real constant.
(i) The line y = x meets y = kex for k  0 at
(A) no point (B) one point
(C) two points (D) more than two points
(ii) The positive value of k for which ke x
x = 0 has only one root is
(A) 1/e (B) 1 (C) e (D) loge2
(iii) For k > 0, the set of all values of k for which ke x x = 0 has two distinct roots is
(A) (0, 1/e) (B) (1/e, 1) (C) (1/e, ) (D) (0, 1)

Match the Column Type Questions

4. Match the column [IIT JEE 2007]
In the following [x] denotes the greatest integer less than or equal to x.
Match the functions in Column I with the properties in Column II.
Column I Column II
(A) x | x | (P) continuous in ( 1, 1)
(B) |x| (Q) differentiable in ( 1, 1)
(C) x + [x] (R) strictly increasing in ( 1, 1)
(D) | x 1|+|x+1| (S) non-differentiable at least at one point in ( 1, 1)

Monotonocity 93
    
5. Let the function g: ( , ) →  − ,  be given by g (u) = 2 tan 1 (eu) . Then, g is:
 2 2 2
(A) even and is strictly increasing in (0,)
(B) odd and is strictly decreasing in ( , )
(C) odd and is strictly increasing in ( , )
(D) neither even nor odd, but is strictly increasing in ( , )

1
6. For the function f(x) = x cos ,x  1, [JEE(Advanced)-2009]
x
(A) for at least one x in the interval [1,), f(x + 2) f(x) < 2
(B) lim f '(x) = 1
x →

(C) for all x in the interval [1, ), f(x + 2) f(x) > 2
(D) f '(x) is strictly decreasing in the interval [1,)

7. The number of distinct real roots of x4 4x3+ 12x2+ x 1 = 0 is [JEE(Advanced)-2011]

b−x
8. Let f: (0,1) →R be defined by f(x) = , where b is a constant such that 0 < b < 1. Then
1 − bx
[JEE(Advanced)-2011]
1
(A) f is not invertible on (0,1) (B) f  f 1 on (0,1) and f '(b) =
f '(0)
1
(C) f = f 1 on (0,1) and f '(b) = (D) f 1is differentiable on (0,1)
f '(0)

9. The number of points in ( , ), for which x2 xsinx cosx = 0, is: [JEE(Advanced)-2013]
(A) 6 (B) 4 (C) 2 (D) 0

10. Let f, g: [ 1, 2] → R be continuous function which are twice differentiable on the interval
( 1, 2). Let the values of f and g at the points 1, 0 and 2 be as given in the following table:
X= 1 X=0 X=2

f(x) 3 6 0

g(x) 0 1 1

In each of the intervals ( 1, 0) and (0, 2) the function (f 3g)'' never vanishes. Then the correct
statement(s) is(are) [JEE(Advanced)-2015]
(A) f '(x) 3g'(x) = 0 has exactly three solutions in ( 1, 0) (0, 2)
(B) f '(x) 3g'(x) = 0 has exactly one solution in ( 1, 0)
(C) f '(x) 3g'(x) = 0 has exactly one solutions in (0, 2)
(D) f '(x) 3g'(x) = 0 has exactly two solutions in ( 1, 0) and exactly two solutions in (0, 2)

94 Monotonocity
 Answer Q. 11, Q.12 and Q.13 by appropriately matching the information given in the three
columns of the following table.
Let f(x) = x + logex x logex, x (0,)
• Column 1 contains information about zeros of f(x), f'(x) and f''(x).
• Column 2 contains information about the limiting behavior of f (x), f'(x) and f''(x) at infinity.
• Column 3 contains information about increasing/decreasing nature of f (x), and f '(x).

Column 1 Column 2 Column 3

(I) f(x) = 0 for some x (1,e2 ) (i) lim f(x) = 0 (P) f is increasing in (0,1)
x →

(II) f ' (x) = 0 for some x  (1, e) (ii) lim f(x) =  (Q) f is decreasing in (e,e2 )
x →

(III) f ' (x) = 0 for some x  (0, 1) (iii) lim f '(x) =  (R) f ' is increasing in (0, 1)
x →

(IV) f " (x) = 0 for some x  (1, e) (iv) lim f "(x) = 0 (S) f ' is decreasing in (e, e2)
x →

11. If f: R → R is a twice differentiable function such that f ''(x) > 0 for all x R, and
 1 1
f   = , f(1) = 1,then [JEE(Advanced)-2017]
2 2
1 1
(A) f '(1) > 1 (B) f '(1) 0 (C) < f '(1)  1 (D) 0 < f ' (1) 
2 2

12. Which of the following options is the only CORRECT combination? [JEE(Advanced)-2017]
(A) (IV) (i) (S) (B) (I) (ii) (R) (C) (III) (iv) (P) (D) (II) (iii) (S)

13. Which of the following options is the only CORRECT combination? [JEE(Advanced)-2017]
(A) (III) (iii) (R) (B) (I) (i) (P) (C) (IV) (iv) (S) (D) (II) (ii) (Q)

14. Which of the following options is the only INCORRECT combination? [JEE(Advanced)-2017]
(A) (II) (iii) (P) (B) (II) (iv) (Q) (C) (I) (iii) (P) (D) (III) (i) (R)

15. For every twice differentiable function f: R → [ 2, 2] with (f(0))2 + (f '(0))2= 85, which of the
following statement(s) is (are) TRUE? [JEE(Advanced)-2018]
(A) There exist r, s R where r < s, such that f is one-one on the open interval (r, s)
(B) There exists x0 ¸( 4, 0) such that |f'(x0)|  1
(C) =
x→

(D) There exists  ( 4,4) such that f () + f"() = 0 and f'()  0

Monotonocity 95
16. Let f : → be defined by [JEE(Advanced)-2021]
x2 − 3x − 6
f(x) =
x2 + 2x + 4
Then which of the following statements is (are) TRUE ?
(A) f is decreasing in the interval ( 2, 1)
(B) f is increasing in the interval (1, 2)
(C) f is onto
 3 
(D) Range of f is − ,2
 2 

17. Let [JEE(Advanced)-2022]



=  sin
k=1
2k
 
6
Let g ∶ [0, 1] → be the function defined by
g(x) = 2x + 2(1 x)

Then, which of the following statements is/are TRUE?

7
(A) The minimum value of g(x) is 26

1
(B) The maximum value of g(x) is 1+2 3

(C) The function g(x) attains its maximum at more than one point
(D) The function g(x) attains its minimum at more than one point

ANSWER KEY

1. (D) 2. (B) 3. (i)-(B) (ii)-(A) (iii)-(A)

4. (A)-(PQR); (B)- (PS); (C)-(RS); (D)-(PQ)

5. (C) 6. (BCD) 7. 2 8. (A) 9. (C)

10. (BC) 11. (A) 12. (D) 13. (D) 14. (D)

15. (ABD) 16. (AB) 17. (ABC)

96 Monotonocity