Chapter 7.1, Problem 60E

(a)

To determine

Prove that the graph of $e^x$ is concave up over every interval of x values.

(b)

To determine

Prove that $e^{\left(\ln a+\ln b\right)/2}\cdot \left(\ln b-\ln a\right)<{\displaystyle \underset{\ln a}{\overset{\ln b}{\int }}{e}^{x}dx}<\frac{e^{\ln a}+e^{\ln b}}{2}\left(\ln b-\ln a\right)$.

(c)

To determine

Prove that $\sqrt{ab}<\frac{b-a}{\ln b-\ln a}<\frac{a+b}{2}$.