Question

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

• A. Differentiable at x=0 if a =0 and b = 1
• B. Differentiable at x=1 if a =1 and b = 0
• C. NOT Differentiable at x=0 if a =1 and b = 0
• D. NOT Differentiable at x=1 if a =1 and b = 1

Answer 1

Answer: (A), (B)

The correct options are
A Differentiable at x=0 if a =0 and b = 1
B Differentiable at x=1 if a =1 and b = 0

$f\left(x\right)=a\cos (|x^3-x|)+b|x|\sin (|x^3+x|)$

For (A) $\to$ If $a=0$, $b=1$, $f\left(x\right)=|x|\sin (|x^3+x|)$

$\Rightarrow$ $f\left(x\right)=x\sin (x^3+x),\forall x\in R$

Hence $f\left(x\right)$ is differentiable.

For (B), (C) $\to$ If $a=1$, $b=0$, $f\left(x\right)=\cos (|x^3-x|)$

$\Rightarrow$ $f\left(x\right)=\cos (x^3-x)$, which is differentiable at $x=1$ and $x=0$

For (D) $\to$ If $a=1$, $b=1$, $f\left(x\right)=\cos (x^3-x)+|x|\sin (|x^3+x|)=\cos (x^3-x)+x\sin (x^3+x)$, which is differentiable at $x=1$