Question

Example 3 find the local maximum and minimum values and saddle points of f(x, y) = x4 + y4 − 4xy + 1. solution we first locate the critical points:

Answer 1

$f(x,y)=x^4+y^4-4xy+1$

$\begin{cases}f_x=4x^3-4y\\f_y=4y^3-4x\end{cases}$

We have critical points whenever both partial derivatives vanish:

$4x^3-4y=0\implies x^3=y\implies x=y^{1/3}$
$4y^3-4x=0\implies y^3=x=y^{1/3}\implies y^6=y\implies y=0,y=-1,y=1$
$\begin{cases}y=0\implies x=0\\y=-1\implies x=-1\\y=1\implies x=1\end{cases}$

The Hessian is

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}12x^2&-4\\-4&12y^2\end{bmatrix}$
$\det\mathbf H(x,y)=(12xy)^2-16$

At (0,0), we have $\det\mathbf H(0,0)=-16$, so (0,0) is a saddle point.

At (-1, -1), we have $\det\mathbf H(-1,-1)=128>0$ and $f_{xx}(-1,-1)=12>0$, so (-1, -1) is a local minimum.

At (1, 1), we have $\det\mathbf H(1,1)=128>0$ and $f_{xx}(1,1)=12>0$, so (1, 1) is also a local minimum.