Chapter 4.3, Problem 63E

The Fresnel function S was defined in Example 3 and graphed in Figures 7 and 8.

(a) At what values of x does this function have local maximum values?

(b) On what intervals is the function concave upward?

(c) Use a graph to solve the following equation correct to two decimal places:

${\displaystyle {\int }_0^x\sin \left(\pi t^2/2\right)}dt=0.2$

EXAMPLE 3 Although a formula of the form $g\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)}$ may seem like a strange way of defining a function, books on physics, chemistry, and statistics are full of such functions. For instance, the Fresnel function

$S\left(x\right)={\displaystyle {\int }_0^x\sin \left(\pi t^2/2\right)}dt$