JEE Advanced 2016 · Paper 2 · Mathematics Q44

MathematicsMultiple correct+4 / -2Question 44 of 54

Let $a,b\in\mathbb R$ and $f:\mathbb R\to\mathbb R$ be defined by $f(x)=a\cos(|x^3-x|)+b|x|\sin(|x^3+x|)$ . Then $f$ is

differentiable at

x=0

a=0

and

b=1

differentiable at

x=1

a=1

and

b=0

NOT differentiable at

x=0

a=1

and

b=0

NOT differentiable at

x=1

a=1

and

b=1

Answer and worked solution

Official answer: A, B

(A) $a=0$ , $b=1$ : $f=|x|\sin(|x|(1+x^2))$ near 0. This is even and $O(x^2)$ , so $f'(0)=0$ exists. True.
(B) $a=1$ , $b=0$ : $\cos|x^3-x|=\cos(x^3-x)$ , which is smooth everywhere. True.
(C) By (B), it is differentiable at 0. False.
(D) Near $x=1$ , $|x|\sin|x^3+x|=x\sin(x^3+x)$ , which is smooth. With the cosine term also smooth, $f$ is differentiable at 1. False.

(A), (B)

Topics:Differentiability

JEE Advanced 2016 · Paper 2, Mathematics question 44