You can use the fundamental theorem of calculus to find f'(x) without having to integrate.
d/(dx)( integral_x^(x^2) e^(-t^2) dt) = 2 e^(-x^4) x - e^(-x^2)
this right hand side can be simplified as e^(-x^4 - x^2) (2 e^(x^2) x - e^(x^4))
which has zeros at approximately .430 and 1.258 I would use a calculator or wolfram alpha to find the zeros.
Please ask me if you do not understand how to use the fundamental theorem of calculus here