The Fresnel function $S$, which has been used in optics and highway design, is defined as $$S(x)=\int_{0}^{\,x} {\sin(\pi t^2/2)\,dt}$$

(a) Determine all values of $x$ between -3 and 3 where this function has a local maximum value.

List all of these values in increasing order. If any blanks are unused, type an upper-case "N" in the corresponding blank.

Local maximum 1: $x$ =

Local maximum 2: $x$ =

Local maximum 3: $x$ =

Local maximum 4: $x$ =

Local maximum 5: $x$ =

Local maximum 6: $x$ =

(b) On what intervals is the function concave upward? There may be more than one correct answer -- you must choose all of the correct intervals to receive full credit on this part.

A. $(\sqrt{4n-1},\sqrt{4n+1}),$ where $n$ is a positive integer.
B. $(-2,0)$
C. $(-\sqrt{4n+2},-\sqrt{4n}),$ where $n$ is a positive integer.
D. $(0,1)$
E. $(-1,0)$
F. $(\sqrt{4n},\sqrt{4n+2}),$ where $n$ is a positive integer.
G. $(-\sqrt{4n-1},-\sqrt{4n-3}),$ where $n$ is a positive integer.
H. $(0,2)$

(c) Use a graph to solve the following equation for $x$, correct to two decimal places: $$\int_{0}^{\,x} {\sin(\pi t^2/2)}\,dt=0.2$$

$x \approx$