Chapter 4.1, Problem 45ES

If $f$ is increasing on an interval $\left[0,b\right),$ then it follows from Definition $\mathrm{4.1.1}$ that $f\left(0\right)<f\left(x\right)$ for each $x$ in the interval $\left(0,b\right)$ . Use this result in these exercises.

(a) Show that $\ln \left(x+1\right)\le x$ if $x\ge 0$ .

(b) Show that $\ln \left(x+1\right)\ge x-\frac{1}{2}{x}^2$ if $x\ge 0$ .

(c) Confirm the inequalities in part (a) and (b) with graphing utility.