Determining the Concavity of \(f(x)=e^x\)

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.