MONOTONOCITY

EXERCISE–I
Q.1 Find the intervals of monotonocity for the following functions & represent your solution set on the number line.
2
(a) f(x) = 2. e x 4x (b) f(x) = ex/x (c) f(x) = x2 ex (d) f (x) = 2x2 – ln | x |
Also plot the graphs in each case.

Q.2 Let f (x) = 1 – x – x3. Find all real values of x satisfying the inequality, 1 – f (x) – f 3(x) > f (1 – 5x)

Q.3 Find the intervals of monotonocity of the function

Q.4 Show that, x3  3x2  9 x + 20 is positive for all values of x > 4.

max {f (t ) : 0  t  x} , 0  x  1
Q.5 Let f (x) = x3  x2 + x + 1 and g(x) = 
 3x ,1  x  2
Discuss the conti. & differentiability of g(x) in the interval (0,2).

Q.6 Find the set of all values of the parameter 'a' for which the function,
f(x) = sin 2x – 8(a + 1)sin x + (4a2 + 8a – 14)x increases for all x  R and has no critical points
for all x  R.
Q.7 Find the greatest & the least values of the following functions in the given interval if they exist.
x  1 
(a) f (x) = sin1  ln x in  , 3  (b) y = xx in (0, ) (c) y = x5 – 5x4 + 5x3 + 1 in [ 1, 2]
2
x 1  3 
1
Q.8 Find the values of 'a' for which the function f(x) = sin x  a sin2x  sin3x + 2ax increases throughout the
3
number line.
ex
Q.9  
Prove that f (x) =  9 cos 2 (2 ln t )  25 cos(2 ln t )  17 dt is always an increasing function of x,  xR
2

 a 2 1
Q.10 If f(x) =   x3 + (a - 1) x2 + 2x + 1 is monotonic increasing for every x  R then find the range of

 3 
values of ‘a’.
Q.11 Find the set of values of 'a' for which the function,
 21  4 a  a 2  3
f(x) =  1  x + 5x + 7 is increasing at every point of its domain.
a 1 
 
Q.12 Find the intervals in which the function f (x) = 3 cos4 x + 10 cos3 x + 6 cos2 x – 3, 0  x  ; is
monotonically increasing or decreasing.
Q.13 Find the range of values of 'a' for which the function f (x) = x3 + (2a + 3)x2 + 3(2a + 1)x + 5 is monotonic
in R. Hence find the set of values of 'a' for which f (x) in invertible.
Q.14 Find the value of x > 1 for which the function

P. No. 8
 x2
F (x) =
1  t  1  is increasing and decreasing.
 t ln  32 
 dt
x

Q.15 Find all the values of the parameter 'a' for which the function ;
f(x) = 8ax  a sin 6x  7x  sin 5x increases & has no critical points for all x  R.
Q.16 If f (x) = 2ex – ae–x + (2a + 1)x  3 monotonically increases for every x  R then find the range of values
of ‘a’.
x2  9 2
Q.17 Construct the graph of the function f (x) =  x and comment upon the following
x 3 x 1
(a) Range of the function,
(b) Intervals of monotonocity,
(c) Point(s) where f is continuous but not diffrentiable,
(d) Point(s) where f fails to be continuous and nature of discontinuity.
(e) Gradient of the curve where f crosses the axis of y.
Q.18 Prove that, x2 – 1 > 2x ln x > 4(x – 1) – 2 ln x for x > 1.
 3 
Q.19 Prove that tan2x + 6 ln secx + 2cos x + 4 > 6 sec x for x   , 2 .
 2 
Q.20 If ax² + (b/x)  c for all positive x where a > 0 & b > 0 then show that 27ab2  4c3.
Q.21 If 0 < x < 1 prove that y = x ln x – (x²/2) + (1/2) is a function such that d2y/dx2 > 0. Deduce
that x ln x > (x2/2)  (1/2).
Q.22 Prove that 0 < x. sin x  (1/2) sin² x < (1/2) ( 1) for 0 < x < /2.
Q.23 Show that x² > (1 + x) [ln(1 + x)]2  x > 0.
Q.24 Find the set of values of x for which the inequality ln (1 + x) > x/(1 + x) is valid.
Q.25 If b > a, find the minimum value of (x  a)3+ (x  b)3, x  R.
EXERCISE–II
Q.1 Verify Rolles throrem for f(x) = (x  a)m (x  b)n on [a, b] ; m, n being positive integer.
Q.2 Let f (x) = 4x3  3x2  2x + 1, use Rolle's theorem to prove that there exist c, 0< c <1 such that f(c) = 0.
Q.3 Let f and g be functions, continuous in [a, b] and differentiable on [a, b]. If f (a) = f(b) = 0 then show
that there is a point c  (a, b) such that g ' (c) f (c) + f '(c) = 0.
Q.4 Assume that f is continuous on [a, b], a > 0 and differentiable on an open interval (a, b).
f (a ) f (b)
Show that if = , then there exist x0  (a, b) such that x0 f '(x0) = f (x0).
a b
Q.5 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).
Q.6 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)
 
Q.7 Using LMVT prove that : (a) tan x > x in  0,  , (b) sin x < x for x > 0
 2
P. No. 9
 3 x 0
 2
Q.8 For what value of a, m and b does the function f (x) =   x  3x  a 0  x 1
 mx  b 1 x  2
satisfy the hypothesis of the mean value theorem for the interval [0, 2].

Q.9 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.

Q.10 Let f defined on [0, 1] be a twice differentiable function such that, | f " (x) |  1 for all x  [0, 1]
If f (0) = f (1), then show that, | f ' (x) | < 1 for all x  [0, 1]
Q.11 Let f be a twice differentiable function on [0, 2] such that f (0) = 0, f (1) = 2, f (2) = 4, then prove that
(a) f '() = 2 for some   (0, 1) (b) f '() = 2 for some   (1, 2)
(c) f "() = 0 for some   (0, 2)
Q.12 Let f be continuous on [a, b] and assume the second derivative f " exists on (a, b). Suppose that the
graph of f and the line segment joining the point a, f (a )  and b, f (b)  intersect at a point
x 0 , f (x 0 )  where a < x0 < b. Show that there exists a point c  (a, b) such that f "(c) = 0.
Q.13 If f is a continuous function on the interval [a, b] and there exists some c  (a, b) then prove that
b

 f ( x) dx = f (c) (b – a).
a
Q.14 Let f (x) be a continuous function on [1, 3], differentiable on (1, 3), f(1) = 5, f(3) = 9 and f '(x)  2
x
for all x in (1, 3). If g(x) =  f ( t ) dt  x  [1, 3], then find the sum of greatest and least value of g(x)
1
on [1, 3].
Q.15 Let f be a differentiable function for all x and that | f ' (x) |  2 for all x. If f (1) = 2 and
f (4) = 8 then compute the value of f 2(2) + f 2 (3).
Q.16 Let f : [0, 8]  R be differentiable function such that f (0) = 0, f (4) = 1, f (8) = 1 then prove that:
1
(a) There exist some c1  (0, 8) where f ' (c1) = .
4
1
(b) There exist some c  (0, 8) where f ' (c) =
12
(c) There exist c1, c2  [0, 8] where 8 f ' (c1) f (c2) = 1.
8
(d) There exist some ,   (0, 2) such that  f ( t ) dt 
= 3  2 f ( 3 )  2 f (3 ) 
0
ANSWER KEY
EXERCISE–I
Q.1 (a) I in (2 , ) & D in ( , 2) (b) I in (1 , ) & D in (  , 0)  (0 , 1)
(c) I in (0, 2) & D in ( , )  (2 , )
1 1 1 1
(d) I for x > or  < x < 0 & D for x <  or 0 < x <
2 2 2 2
Q.2 (–2, 0)  (2, )
Q.3 (a) I in [0, 3/4)  (7/4 , 2 ] & D in (3/4 , 7 /4)
(b) I in [0 , /6)  (/2 , 5/6)  (3/2 , 2 ] & D in (/6 , /2)  (5/6, 3 /2)]
Q.5 continuous but not diff. at x = 1 Q.6 
a <  2 5  or a > 5

P. No. 10
Q.7 (a) (/6)+(1/2)ln 3, (/3) – (1/2)ln 3, (b) least value is equal to (1/e)1/e, no greatest value, (c) 2 & 10
Q.8 [1, ) Q.10 a  (– , – 3]  [1 , ) Q.11 [ 7,  1)  [2, 3]
Q.12 increasing in x  (/2 , 2/3) & decreasing in [0 , /2)  (2/3 , ]
3
Q.13 0a Q.14  in (3, ) and  in (1, 3) Q.15 (6, ) Q.16 a0
2
EXERCISE–II
mb  na
Q.1 c= which lies between a & b Q.8 a = 3, b = 4 and m = 1
mn
Q.14 14 Q.15 52exercise–ii
mb  na
Q.1 c= which lies between a & b Q.8 a = 3, b = 4 and m = 1
mn
Q.14 14 Q.15 52

P. No. 11