Chapter 4.5, Problem 87E

(a)

To determine

To graph: The function $f\left(x\right)=6\sin x{\cos }^{2}x$ and $g\left(t\right)={\displaystyle {\int }_0^{t}f\left(x\right)dx}$ on same plot.

(b)

To determine

The explanation of non-negative g.

(c)

To determine

The points on graph of g depicting the extrema of f.

(d)

To determine

Whether the zeros of f correspond to an extremum of g.

(e)

To determine

To graph: The function $h\left(t\right)={\displaystyle {\int }_{\frac{\pi }{2}}^{t}f\left(x\right)dx}$ and find the relationship between $g$ and $h$.

To determine

To calculate: The relationship between $g$ and $h$.