Locate and classify stationary points

Question

Locate and classify as maxima, minima or saddle point the stationary points of the surface given by the equation $$z=(5x+7y-25)e^{-(x^2+xy+y^2)}.$$

Stationary points are the points where the gradient vector is zero.

Answer 1

Find the first and second partial derivatives. Set both first partial derivatives equal to zero, and solve as a system of equations. Each solution to this system of equations is a stationary point. Then, apply the second derivative test to all stationary points. The second derivative test is as follows: Consider the second partial derivatives of a function $f$. If the expression $(d^2f/dx^2)$ * $(d^2f/dx^2)$ - $(d^2f/dxdy)^2$ $>0$ and $d^2f/dx^2$ $>0$ at a stationary point, that stationary point is a local minimum of $f$. If the expression $d^2f/dx^2$ * $d^2f/dx^2$ - $(d^2f/dxdy)^2$ $>0$ and $d^2f/dx^2$ $<0$ at a stationary point, that stationary point is a local maximum of $f$. Finally, if the expression $d^2f/dx^2$ * $d^2f/dx^2$ - $(d^2f/dxdy)^2$ $<0$ at a stationary point, that stationary point is a saddle point.