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

Oct 31, 2015

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

#### Explanation:

The partial derivatives are:

$\frac{\partial f}{\partial x} = 3 {x}^{2} {e}^{{x}^{3}} - y {e}^{x y}$ and $\frac{\partial f}{\partial y} = 3 {y}^{2} - x {e}^{x y}$.

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

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.