Figure 39 $f(x) = e^x$ is concave up on its domain.

Show that $f(x)=e^{x}$ is concave up on its domain.

Solution The domain of $f(x) =e^{x}$ is all real numbers. The first and second derivatives of $f$ are $$f^\prime (x)=e^{x}\qquad f^{\prime \prime} (x)=e^{x}$$

Since $f^{\prime \prime} (x) >0$ for all real numbers, by the Test for Concavity, $f$ is concave up on its domain.