4.18 Solution to geometric example

To find the minimum of $S(x,y)$, first we calculate partial derivatives:

Therefore the stationary points are given by $x=\,{\frac{64}{y^{2}}}$ and $y=\,{\frac{64}{x^{2}},}$ so $x=\,{64\cdot\frac{x^{4}}{64^{2}}=}\,{\frac{x^{4}}{64}.}$ It follows that $x^{3}=64$, so $x=4$ and hence $y=\frac{64}{4^{2}}=4$. Thus there is only one stationary point: $P=(4,4)$.

Now the second order partial derivatives at $P$ are: $S_{xx}=\,{2,}$ $S_{xy}=\,{1}$ and $S_{yy}=\,{2.}$ Then $\Delta=\,{3}$ and so $P$ is a local minimum.

In fact $P$ is the overall minimum (for $x,y>0$) and so it gives us the minimum value of $S$, which is $48$.