Section 8.2 A Saddle Point Example

The partial derivatives exist everywhere, so the only point at which a local extremum can occur is the point \((0,0)\text{.}\) Note that along the \(x\)-axis the function takes negative values, \(f(x,0) = -x^2/3 \lt 0\text{,}\) whereas along the \(y\)-axis the function takes positive values, \(f(0,y) = y^2/3 \gt 0\text{,}\) as shown in Figure 8.11. Hence inside every open disk around \((0,0)\text{,}\) there are points \((x,y)\) such that \(f(x,y) \gt f(0,0)\) and also there are points such that \(f(x,y) \lt f(0,0)\text{,}\) which means \((0,0)\) is a saddle point. Note that in this example the two partial derivatives are zero at \((0,0)\text{;}\) however, the function does not have a local extremum at this point.