Chapter 4.6, Problem 82P

a.

To determine

To argue: that $f$ has a maximum value $M$ and a minimum value $m$ on the interval $\left[a,b\right]$ .

b.

To determine

To argue: that $m\left(b-a\right)\le {\displaystyle {\int }_a^{b}f\left(x\right)dx\le M\left(b-a\right)}$ and thus that the average value of $f$ on $\left[a,b\right]$ is between $m$ and $M$ .

c.

To determine

To argue: That there is some point $c\in \left(a,b\right)$ for which $f\left(c\right)$ is equal to the average value of $f$ on $\left[a,b\right]$ .