A critical value c of a function f(x) is a value where f'(c) = 0 or f'(c) is undefined.
The challenge here is to find f'(x) as we do not have a simple rule to handle |x| in differentiation. We could break it into a piecewise function split at x = 0, but let's use another trick in this case.
For all x in RR we have |x| = sqrt(x^2). Thus we can rewrite the function as
f(x) = sin(sqrt(x^2))
Then, applying the chain rule, we have
f'(x) = d/dx sin(sqrt(x^2))
= cos(sqrt(x^2))(d/dxsqrt(x^2))
= cos(sqrt(x^2))(1/(2sqrt(x^2)))(d/dxx^2)
= cos(sqrt(x^2))(1/(2sqrt(x^2)))(2x)
= (xcos(|x|))/|x|
Then, f'(x) is undefined when x=0, and
f'(x) = 0 <=> cos(|x|) = 0
<=> x = pi/2 + npi where n in ZZ
Thus the critical values of sin|x| are
{pi/2 + npi|n in ZZ} uu {0}
