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}#
