Tutorial Sheet-1 AUTUMN 2019

MATHEMATICS-I(MA10001) August 1,2019

1. Verify which functions satisfy the conditions of the Rolle’s theorem and if satisfies
find c which satisfy the conclusion of the Rolle’s theorem:
(a) f (x) = x2 + cos(x) on [ −π
4 4
, π] (b)f (x) = 1 − |x − 1| on [0, 2]
1 1 1 2
(c) f (x) = sin( x ) on [ 3π , 2π ] (d)f (x) = 1 − (x − 1) 3 on [0, 2].

2. Calculate ξ ∈ (a, b) in cauchy MVT for each of the following pairs:

(a) f (x) = sin x, g(x) = cos x on [ π4 , 3π ],
3 √ 4
(b) f (x) = (1 + x) 2 , g(x) = 1 + x on [0, 21 ].

3. Prove that between any two real roots of the equation ex sin x + 1 = 0 there is
atleast one real root of the equation tan x + 1 = 0.

4. Show that the formula in the Lagrange MVT can be written as follows:

f (x + h) − f (x)
= f 0 (x + θh)
h
where 0 < θ < 1. Determine θ as a function of x and h when
(a)f (x) = x2 (b)f (x) = ex (c)f (x) = log x , x > 0.
Keep x 6= 0 fixed, and find lim θ in each case.
h→0

5. Let f be a function having a finite derivative f 0 in the half-open interval 0 < x ≤ 1

such that |f 0 (x)| < 1. Define an = f ( n1 ) for n = 1, 2, 3, .....Show that lim an exists.
n→∞

6. Assume f has a finite derivative in (a, b) and is continuous on [a, b] with f (a) =
f (b) = 0. Prove that for every real λ there is some c in (a, b) such that f 0 (c) = λf (c).

7. Answer the followings:

(a) Suppose, f (x) is continuous on [−7, 0] and differentiable in (−7, 0) such that
f (−7) = −3 and |f 0 (x)| ≤ 2. Then, what is largest possible value of f (0).
√
(b) Use Lagrange MVT to estimate 3 28.
(c) If f 00 (x) ≥ 0 on [a, b] prove that f x1 +x ≤ 21 f (x1 ) + f (x2 ) for any two

 
2
2
points x1 and x2 in [a, b].

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 8. If f has a finite third derivative f 000 in [a, b] and if f (a) = f (b) = f 0 (a) = f 0 (b) = 0.
Prove that f 000 (c) = 0 for some c in (a, b).

9. Prove that
2x
(a) π
< sin x < x for 0 < x < π2 .
(b) nan−1 (b − a) < bn − an < nbn−1 (b − a) where 0 < a < b and n > 1.
x
(c) 1+x
< log(1 + x) < x for all x > 0.

10. Use Rolle’s theorem to prove the following:

R2
(a) Let f : [1, 3] → R be a continuous function such that 1 f (x)dx = 2 and
R3
1
f (x)dx = 3. Then show that there exist some c ∈ (2, 3) such that
Z c
cf (c) = f (x)dx.
1

(b) Let f : [a, b] → R be a continuous function on [a, b] and f 00 (x) exists for all
x ∈ (a, b). Let a < c < b, then there exists a point ξ in (a, b) such that
b−c c−a 1
f (c) = f (a) + f (b) + (c − a)(c − b)f 00 (ξ).
b−a b−a 2

11. If c0 + c21 + c32 + .... + n+1

cn
= 0 where c0 , c1 , ..., cn are real. Show that the equation
n
c0 + c1 x + ..... + cn x = 0 has atleast one real root between 0 and 1.

12. Answer the followings:

(a) Assume f is continuous on [a, b] and has a finite second derivative f 00 in the
open interval (a, b). Assume that the line segment joining the points A =
(a, f (a)) and B = (b, f (b)) intersects the graph of f in a third point P different
from A and B. Prove that f 00 (c) = 0 for some c in (a, b).
(b) If f is differentiable on [0, 1] show by Cauchy’s MVT that the equation f (1) −
0 (x)
f (0) = f 2x has atleast one solution in (0, 1).
(c) 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 and c2 in (a, b) such that f 0 (c1 ) + f 0 (c2 ) = 2.

13. Answer the followings:

(a) If f (x) and φ(x) are continuous on [a, b] and differentiable on (a, b), then show
that
f (a) f (b) f (b) f 0 (c)
= (b − a) , a < c < b.
φ(a) φ(b) φ(b) φ0 (c)
(b) Let f be continuous on [a, b] and differentiable on(a, b). Using Cauchy’s MVT
show that if a ≥ 0 then there exist x1 , x2 , x3 ∈ (a, b) such that

f 0 (x2 ) f 0 (x3 )
f 0 (x1 ) = (b + a) = (b2 + ba + a2 ) .
2x2 3x23

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14. Use CMVT to prove the followings:
x2
(a) Show that 1 − 2!
< cos x for x 6= 0.
(b) Let f be continuous on [a, b], a > 0 and differentiable on (a, b). Prove that
2 2 f (b)
there exist c ∈ (a, b) such that b f (a)−a
2
b −a2 = 12 [2cf (c) − c2 f 0 (c)].
√
2 ln x 1−x2
(c) Show that 2 arcsin x−π
< x
.