Chapter 5.8, Problem 79E

To determine

To prove: $e^x\ge 1+x$, with the help of property below.

If $f(t)\ge g(t)$ for $t\ge 0$ then for all $x\ge 0$, ${\displaystyle \underset{0}{\overset{x}{\int }}f\left(t\right)dt}\ge {\displaystyle \underset{0}{\overset{x}{\int }}g\left(t\right)dt}$

Then prove, by successive integration the following inequalities

$e^x\ge 1+x+\frac{x^2}{2}$ and $e^x\ge 1+x+\frac{x^2}{2}+\frac{x^3}{6}$