Question

Let $a,bϵR$ and $f:R\to R$ be defined by $f\left(x\right)=acos(|x^3-x|)+b|x|sin(|x^3+x|)$.
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. Differentiable at x=0 if a =1 b=0
• D. Differentiable at x=1 if a =1 b=1

Answer 1

Answer: (A), (B), (C), (D)

Differentiable at x=1 if a =1 b=1

$f\left(x\right)=acos(|x^3-x|)+b|x|sin(|x^3+x|)$
(a) If $a=0,b=1$
$\Rightarrow f\left(x\right)=|x|sin|x^3+x|$
$=xsin(x^3+x),xϵR$
$\therefore$ f is differentiable every where.

(b) , (C) If $a=1,b=0\Rightarrow f\left(x\right)=co{s}^3(|x^3-x|)=co{s}^3(x^3-x)$
Which is differentiable every where.

(d) When $a=1,b=1,f\left(x\right)=cos(x^3-x)+xsin(x^3+x)$
Which is differentiable at x=2
$\therefore$ All options are correct.