(Solved): (5) Consider the following Fresnel integral used in the field of optics. \[ C(x)=\int_{0}^{x} \cos ...

(5) Consider the following Fresnel integral used in the field of optics. \[ C(x)=\int_{0}^{x} \cos \left(\frac{\pi t^{2}}{2}\right) d t . \] (a) Find \( C^{\prime}(x) \). (b) Find \( C^{\prime \prime}(x) \). (c) Does \( C \) have a local maximum value at \( x=2 \) ? Explain. (d) Use the Intermediate Value Theorem to show that \( C(x) \) has an inflection point on the interval \( (1, \sqrt{3}) \).