# 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#?

##### 2 Answers

The unique critical point is

#### Explanation:

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).

Since

See Explanation

#### Explanation:

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

Calculate

Apply the second derivative test:

Since

To conclude: