What are the critical points of #f(x,y) = e^(x^3) + y^3 - e^(xy)#?

1 Answer
Oct 31, 2015

Answer:

The critical points are #(x,y)=(0,0)# and #(x,y) approx (0.3705, 0.3768)#. The first one is a saddle point and the second one is a local minimum.

Explanation:

The partial derivatives are:

#(partial f)/(partial x)=3x^2e^(x^3)-ye^(xy)# and #(partial f)/(partial y)=3y^2-xe^(xy)#.

If you set these both equal to 0, the resulting system of equations clearly has #(x,y)=(0,0)# as a solution. You must use a calculator or software to approximate the other solution, and it turns out to be #(x,y) approx (0.3705, 0.3768)#.

The first one is a saddle point and the second one is a local minimum.

Below is a 3-dimensional picture of the graph of this function. The positive #x#-axis points to the right and the positive #y#-axis points into the screen.

enter image source here