Indian Institute of Technology Bhubaneswar
School of Basic Sciences
Mathematics-1 (MA1L001) (Autumn-2019)
Assignment-1
1. Use ǫ-δ definition of a “Limit" to show the following:
Let f and g be functions and both lim f (x) and lim g(x) exist and are
x→a x→a
finite. Then,
a) lim (f (x) + g(x)) = lim f (x)+lim g(x), b) lim (f (x)g(x)) = lim f (x) lim g(x).
x→a x→a x→a x→a x→a x→a
1 1
c) lim = , provided lim g(x) 6= 0.
x→a g(x) lim g(x) x→a
x→a
1
2. By using ǫ-δ definition of a “Limit" show that lim sin does not exist.
x→0 x
3. Discuss the applicability of Rolle’s theorem for the following function:
(i) f (x) = tan x, ∀ x ∈ [0, π]
(ii) f (x) = 1 − |x − 1|, ∀ x ∈ [0, 2]
(iii) f (x) = |x|, ∀ x ∈ [−1, 1].
4. Show that the equation 3x + 2 cos x + 5 = 0 has exactly one real root.
(Use IVT and Rolle’s theorem).
5. Prove that the equation (x − 1)3 + (x − 2)3 + (x − 3)3 + (x − 4)3 = 0
has only one positive real root.
6. Show that the equation x13 + 7x3 − 5 = 0 has exactly one real root.
7. Let f : [0, 1] −→ R be differentiable function such that |f ′ (x)| < 1 for
all x ∈ [0, 1]. Prove that there exists at most one a ∈ [0, 1] such that
f (a) = a.
8. Use the Mean value theorem (MVT) to establish the following inequal-
ities.
a) ex > 1 + x for x > 0.
√ √
b) for all n ∈ N, show that √1 < n+1− n< 1
√ .
2 n+1 2 n
c) | sin x − sin y| ≤ |x − y| for all x ∈ R.
9. Does there exist a differentiable function f : [0, 2] −→ R satisfying
f (0) = −1, f (2) = 4 and f ′ (x) ≤ 2 for all x ∈ [0, 2].
10. Let f : [0, 1] −→ R be twice differentiable. Suppose that the line
segment joining the points (0, f (0)) and (1, f (1)) intersects the graph
of f at a point (a, f (a)) where 0 < a < 1. Show that there exists
x0 ∈ [0, 1] such that f ′′ (x0 ) = 0.
11. Let f : [a, b] −→ R be differentiable. Then f is constant if and only if
f ′ (x) = 0 for every x ∈ [a, b].
12. Let f and g be functions, continuous on [a, b], differentiable on (a, b)
and let f (a) = f (b) = 0. Prove that there is a point c ∈ (a, b)
such that g ′(c)f (c) + f ′ (c) = 0 (Hint: Consider the function F (x) =
f (x) exp(g(x)) and use Rolle’s Theorem).
13. Suppose f and g are two continuous functions from [a, b] to R. If f ′ (x) =
g ′ (x) at each point x ∈ (a, b), then there exists a constant C such that
f (x) = g(x) + C for all x ∈ (a, b).
2
14. Using Cauchy Mean Value Theorem (CMVT), show that 1 − x2 < cos x
for x 6= 0.
15. Let f : [a, b] −→ [a, b] be a continuous function. Prove that, there
exists a point c ∈ [a, b] such that f (c) = c. [Hint: Consider the function
g(x) = f (x) − x and use IVT]
16. Let f : [a, b] −→ R be a continuous function on [a, b] with f ′ (x) > 0 in
(a, b). Show that f is strictly increasing in [a, b].
17. If c0 + c21 + c32 + · · · + n+1
cn
= 0 for ci ∈ R, show that c0 + c1 x + c2 x2 +
n
· · · + cn x = 0 has at least one real root between 0 and 1.
18. A function f is differentiable and f ′ is continuous on [0, 2] and f (0) =
0, f (1) = 2, f (2) = 1. Prove that f ′ (c) = 0 for some c ∈ (0, 2).
19. Prove that between any two roots of the equation ex cos x+1 = 0, there
is at least one real root of the equation ex sin x + 1 = 0.
20. Show that f (x) = 4x5 + x3 + 7x − 2 has exactly one real root.
21. Suppose f is continuous and differentiable on [6, 15]. Let’s also suppose
f (6) = −2 and f ′ (x) ≤ 10. Find a possible bound for the value f (15)?
(Ans: 88)
22. Let f (t) = At2 + Bt + C, A, B, C ∈ R and A 6= 0, be a function
describing distance traveled by a body in time t, t ∈ [a, b]. Show that
the average speed of the body is always attained at the midpoint a+b2
.
[Hint: Use mean value theorem.]
2
23. Prove that between any two real roots of ex sin x + 1 = 0, there is at
least one real root of tan x + 1 = 0.
24. Let f : [a, b] → R be continuous on [a, b] and f ′′ (x) exists for all
x ∈ (a, b). Let a < c < b. Prove that there exists ξ ∈ (a, b) such that
b−c c−a 1
f (c) = f (a) + f (b) + (c − a)(c − b)f ′′ (ξ).
b−a c−b 2
x
25. Prove that 1+x
< log(1 + x) < x, ∀ x > 0.
26. Using Lagrange Mean Value theorem, Prove that 0 < x1 log( e x−1 ) < 1.
x
27. Let f : [0, 1] → R be continuous and differentiable on (0,1) and
lim f ′ (x) = α, where α ∈ R. Show that f ′ (0) exists and f ′ (0) = α.
x→0
28. Calculate ξ in Cauchy Mean Value Theorem for each of the following
pairs of functions:
(i) f (x) = sin x, g(x) = cos x, x ∈ [π/4, 3π/4]
√
(ii) f (x) = x, g(x) = √1x , x ∈ [1, 3]
(iii) f (x) = log x, g(x) = x1 , x ∈ [1, e]
√
(iv) f (x) = (1 + x)3/2 , g(x) = 1 + x, x ∈ [0, 1/2].
29. If f is differentiable on [0, 1], show by Cauchy Mean Value Theorem
′ (x)
that the equation f (1) − f (0) = f 2x has at least one solution in (0, 1).
30. Let f : [a, b] → R be differentiable and a ≥ 0. Using Cauchy Mean
′ (c )
Value Theorem, show that there exists c1 , c2 ∈ (a, b) such that fa+b 1
=
′
f (c2 )
2c2
.
31. Using CMVT prove that:
If f be continuous on [a, b] and differentiable on (a, b), then ∃ c ∈ (a, b)
such that
bf (a) − af (b)
= f (c) − cf ′ (c).
b−a
************************* All The Best *************************
