Second Partials of x^3 - 3xy^2 and Why They Cancel

The computation and the cancellation

Both first partials are easy polynomials. Treat $y$ as constant for the first and $x$ as constant for the second.

Differentiating each again gives second partials that are all linear.

The pure second partials are exact negatives of each other, so their sum vanishes identically. That equation has a name, and functions that satisfy it are called harmonic.

What the Hessian says about the origin

Setting both first partials to zero gives $3x^{2} = 3y^{2}$ and $xy = 0$, and the only solution is $x = y = 0$. So the origin is the single critical point.

The second derivative test uses the Hessian determinant $D = f_{xx}f_{yy} - \left(f_{xy}\right)^{2}$.

Away from the origin $D < 0$, but at the origin itself $D = 0$ and the test gives no verdict. You have to look at the function directly: along the line $y = 0$ it reduces to $x^{3}$, which is positive for $x > 0$ and negative for $x < 0$, so the origin is neither a maximum nor a minimum.

This surface is the monkey saddle: three directions go down and three go up around the origin, which is why no single sign of curvature describes it.

The mistake students make

The common error is reporting a saddle point at the origin because $D$ happens to be negative nearby. The second derivative test is evaluated at the critical point only, and there $D = 0$, so the honest answer is that the test is inconclusive and you must argue from the function.

• $D > 0$ and $f_{xx} > 0$: local minimum.
• $D > 0$ and $f_{xx} < 0$: local maximum.
• $D < 0$: saddle point.
• $D = 0$: no conclusion, investigate by hand.

The other slip is a sign error in $f_{yy}$. Differentiating $-6xy$ with respect to $y$ gives $-6x$, not $6x$, and getting that wrong destroys the cancellation that makes the function harmonic.