Chapter 5.4, Problem 58E

(a).

To determine

To sketch: The figure to illustrate the geometric meaning of the inequality $\left(b-a\right)f\left(\frac{a+b}{2}\right)\le {\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx\le \left(b-a\right)\frac{f\left(a\right)+f\left(b\right)}{2}}$, assuming the $f$ be non negative and $f^{\text{'}\text{'}}\left(x\right)>0$ on the interval $\left[a,b\right]$.

(b).

To determine

To interpret: The result when the inequality is divided by (b – a) in terms of the average value of $f$ on $\left[a,b\right]$.