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If $f(x)=a|\sin x|+b e^{|x|}+c|x|^3$, where $a, b, c \in R$, is differentiable at $x=0$, then
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$a=0, b=0, c$ is any real number
$a=0, b=0, c$ is any real number
$|\sin x|$ and $e^{|x|}$ are not differentiable at $x=0$ and $|x|^3$ is differentiable at $x=0$. $\therefore$ for $f(x)$ to be differentiable at $x=0$, we must have $a=0, b=0$ and $c$ is any real number.
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