Let the (x; y) coordinates represent locations on the ground. The height h of a particle is governed by the function described below. Classify all the critical
points of height and find the value(s) of height h at those points.
h(x; y) = 8x + 10y - 4xy - 4x^2 - 4y^3

The critical points of h(x,y) occur wherever its partial derivatives and vanish simultaneously. We have

Substitute y in the second equation and solve for x, then for y :

This is to say there are two critical points,

To classify these critical points, we carry out the second partial derivative test. h(x,y) has Hessian

whose determinant is . Now,

• if the Hessian determinant is negative at a given critical point, then you have a saddle point

• if both the determinant and are positive at the point, then it's a local minimum

• if the determinant is positive and is negative, then it's a local maximum

• otherwise the test fails

We have

while

So, we end up with