Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Question

Show that $g(x)$ is bounded below, for $0\leq x$:

a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$

b) $g(x)=e^x$

For (a), $g(x)\geq0$ for $x\geq0$.

For (b), $g(x)\geq1$ for $x\geq0$.

Is that all I have to say? Or is there a more technical definition/proof?

Answer 1

Hint:

For a) you can show that $g(x)$ attains a minimum on $(0, \infty)$ by setting $g'(x) = 0$ and verifying that its critical point is a local minimum.

For b) you can note that $g(0) = 1$ and that $g(x)$ is increasing.