To determine the nature of the stationary point of the function

and sketch its graph.

Solution. We compute the derivative by the product rule

Now the stationary point occurs where $f^{\prime}(x)=0$; that is, $\log x=-1$, so $x=1/e$, and $f(1/e)=-1/e$ and $f^{\prime\prime}(1/e)=e>0.$ Hence there is a local minimum at $(1/e,-1/e).$