If f(x)=a| x|+be^|x|+c|x|^3, where a,b,cR, is differentiable at x=0, then

Question

If $f(x)=a|\sin x|+be^{|x|}+c|x|^3$, where $a,b,c\in\mathbb{R}$, is differentiable at $x=0$, then

Accepted Answer

$|\sin x|$ and $e^{|x|}$ both have corners (non-differentiable) at $x=0$, while $|x|^3$ is differentiable there. For $f$ to be differentiable at $0$, we need $a=0$ and $b=0$, with $c$ arbitrary.

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