INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Course Name: Several variable calculus and Differential geometry Course Code: MTH 305 A

Assignment - 1:

(1) (i) If f is differentiable at c, then show that f is continuous at c.

(ii) Let f : R −→ R satisfies | f(x)− f(y) |≤ λ(x− y)2015 for some real number λ > 0 and x, y ∈ R.

Prove that f is constant function.

(iii) Let f : (a, b) −→ R be a continuous function and c ∈ (a, b). Suppose that f ′ is defined on (a, b)

except possibly at c, and suppose that f ′(x) −→ L as x −→ c. Prove that f ′(c) = L.

(iv) If f(x) = | x |3 for all x ∈ R, compute f ′(x), f′′(x) and show that f

′′′(0) does not exist.

(v) Define f(0) = 0 and for x 6= 0, f(x) = x2sin(1/x). Show that f ′(0) = 0 and limx→0f′(x) does

not exist.

(vi) Let f be defined on an open interval (a, b) and assume that for some point c in (a, b) we have

f ′(c) < 0 or f ′(c) = −∞. Then show that f is decreasing in a neighborhood of the given point

c.

(vii) Give an example of a function f defined on a subset S in R wchich have local maxima at several

points in S without having an absolute maximum on the whole set S.

(2) State and prove the following results:

(i) The chain rule for differentiation of composite functions.

(ii ) Rolle’s theorem

(iii) Mean- value theorem for derivatives

(iv) Generalized Mean-value theorem

(v) Intermediate -value theorem for derivatives

(vi) Taylor’s theorem

(3) If f and g are continuous on [a, b] and have equal finite derivatives in (a, b), then prove that f − g is

constant on [a, b].

(4) Let f be continuous on (a, b) with finite derivative f ′ in (a, b), except possibly at c ∈ (a, b). If

limx→cf′(x) exists then show that f ′(c) must also exists and have same value as the limit.

(5) Suppose 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 number λ there is some point p ∈ (a, b) such that f ′(p) = λf(p).

(6) Assume f is nonnegetive and has finite third order derivative f′′′

in the open interval (0, 1). If

f(x) = 0 for at least two values of x in (0, 1), prove that f′′′

(c) = 0 for some c in (0, 1).

(7) Let N be a fixed positive number. Show that there is no function f satisfying the following three

conditions: f ′(x) exists for x ≥ 0, f ′(0) = 0, f ′(x) ≥ N for x > 0.

(8) Suppose f is differentiable on [a, b], f(a) = 0, and there is a real number A such that

| f ′(x) |≤ A | f(x) | on [a, b]. Prove that f(x) = 0 for all x ∈ [a, b].1

Hint: Fix x0 ∈ [a, b], let M0 = sup | f(x) |, M1 = sup | f ′(x) | for a ≤ x ≤ x0. For any such x,

| f(x) |≤M1(x0 − a) ≤ A(x0 − a)M0. Hence M0 = 0 if A(x0 − a) < 1. That is, f = 0 on [a, x0].

(9) Assume that f has finite derivative in (a, b) and is continuous on [a, b], with a ≤ f(x) ≤ b for all

x ∈ [a, b] and |f ′(x)| ≤ α < 1 for all x ∈ (a, b). Prove that f has a unique fixed point in [a, b].

(10) Let f and g be two functions both taking values in Rn and differentiable at each points of (a, b).

Show that < f, g >′ exists in (a, b) and < f, g >′=< f ′, g > + < f, g′ >. Hence show that if f has

constant norm ||f || then f(t).f ′(t) = 0 on (a, b).

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