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Question

Which of the following functions is differentiable at x=0?

A
cos(|x|)+|x|
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B
cos(|x|)|x|
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C
sin(|x|)+|x|
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D
sin(|x|)|x|
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Solution

The correct option is D sin(|x|)|x|
g(x)=cos(|x|)±|x|=cos(x)±|x| (cos x is an even function)
g(x) is not differentiable at x=0

f(x)=sin(|x|)|x|

f(x)={sin xx, x0sin x+x, x<0}

f(0+)=0=f(0)

So, f(x) is differentiable at x=0.

Similarly, check for sin(|x|)+|x|

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