Examples

Minimum vs. saddle

Minimum vs. saddle

Problem: Consider the function $f$:

$$f(x,y)=(x^2+y^2)e^{-x}.$$

Determine the local extrema of $f$ by finding and analyzing its critical points.

Solution: We set $f_x=0,f_y=0$ and solve this system of equations for points $(x,y)$. By computing the partials, we obtain:

$$\begin{array}{rl}(2x-x^2-y^2)e^{-x}& =0\\ 2y{e}^{-x}& =0.\end{array}$$

The second equation implies $y=0$ because $e^{-x}$ is never zero. Plugging this fact into the first equation gives $x(2-x)e^{-x}=0$ which implies $x=0,2$. So the critical points as given by components are $(0,0)$ and $(2,0)$.

Now let us analyze these critical points using the discriminant. Compute:

$$\begin{array}{rl}{f}_{xx}& =(2-4x+x^2+y^2)e^{-x}\\ f_{yy}& =2e^{-x}\\ f_{xy}& =-2y{e}^{-x}.\end{array}$$

Therefore $D(x,y)=2(2-4x+x^2-y^2)e^{-2x}$, and in particular $D(0,0)=4$, $D(2,0)=-4e^{-4}$. We also need $f_{xx}(0,0)=2>0$ and $f_{xx}(2,0)=-2e^{-2}<0$. The discriminant test concludes that $(0,0)$ is a local minimum, while $(2,0)$ is a saddle point.

3D plot: https://www.math3d.org/cDHxQockR

Also try Desmos3d: https://www.desmos.com/3d/17d5d68086

Minimal distance

Minimal distance

Problem: Consider the surface given by points with coordinates satisfying $y^2=9+xz$. Find the point(s) $(x,y,z)$ closest to the origin.

Solution: Our goal is to extremize the distance to the origin of an arbitrary point on the surface. The distance formula gives $d=\sqrt{x^2+y^2+z^2}$. This function will be extremized at the same place(s) that the function $d^{\prime }=x^2+y^2+z^2$ is extremized because $\sqrt{-}$ is monotone increasing for positive inputs. So we minimize $d^{\prime }$ instead. Next we write the distance as a function of just two variables. Using the surface equation to determine $y^2$, we write $d^{\prime }=9+xz+x^2+z^2$. Then:

$$\begin{array}{rl}{d}_x^{\prime }& =z+2x\\ d_z^{\prime }& =x+2z.\end{array}$$

Setting these simultaneously to zero and solving gives $(x,z)=(0,0)$, and $(x,y,z)$ is $(0,3,0)$ or $(0,-3,0)$. The distance at these points is 3.