I'm just checking my answers to the following question to make sure my reasoning is sound.
I want to find the values of $x$ for where the Fresnel function takes local minimum values. This function being: $\int_{0}^{x} \sin{\frac{\pi t^2}{2}} dt$.
So I used the Fundamental Theorem of Calculus to find the first derivative and from that found the second derivative to be $\pi x \cos{\frac{\pi x^2}{2}}$. I want this to be greater than zero for local minimum values.
I know from looking at the graph of $\cos(x)$ that it is greater than zero for even multiples of $\pi$ so $\frac{x^2}{2}$ must be even i.e. $x = \sqrt{2m}$ where $m$ is an even positive integer?
Is this correct?