How do you find the critical point and determine whether it is a local maximum, local minimum, or neither for #f(x, y) = x^2 + 4x + y^2#?
The unique critical point is
The first-order partial derivatives of
Setting both of these equal to zero results in a system of equations whose unique solution is clearly
The second-order partials are
This makes the discriminant for the (multivariable) Second Derivative Test equal to
which means the critical point is either a local max or a local min (it's not a saddle point).
The critical point makes both partial derivatives
For this function there is one critical point:
To determine whether
Evaluate the second partials at the critical point (In this case they are all constant, but in general we cannot skip this step.)
At the critical point
Apply the second derivative test: