10. Let f(x,y) = x³ + 6xy + y³ + 3. What are all the points at which f has a relative maximum?

First find the points (x,y) where both the partial derivatives with respect to x and y of f(x,y) are equal to zero. Then determine whether f(x,y) has a relative maximum at each point by using the second derivative test.

Set the partial derivatives with respect to x and y of f(x,y) equal to zero,

By observation, the only two points that satisfy the above equations are (0,0) and (-2,-2).

The second partial derivatives of f(x,y) are:

The only critical point demonstrates that the second partial derivatives are negative is (-2,-2). Hence by the second derivative test, f(x,y) has a local maximum at (-2,-2).