# 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

As a result, the Second Derivative Test's discriminant (multivariable) equals

which indicates that the critical point is not a saddle point but rather either a local max or a local min.

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See Explanation

At the critical point, evaluate the second partials (they are all constant in this case, but generally we can't skip this step).

Use the test for the second derivative:

To sum up:

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To find the critical point, we first need to find the partial derivatives of the function ( f(x, y) ):

( \frac{{\partial f}}{{\partial x}} = 2x + 4 )

( \frac{{\partial f}}{{\partial y}} = 2y )

Setting both partial derivatives equal to zero to find critical points:

( 2x + 4 = 0 \Rightarrow x = -2 )

( 2y = 0 \Rightarrow y = 0 )

So, the critical point is ( (-2, 0) ).

To determine whether it's a local maximum, local minimum, or neither, we use the second partial derivative test.

( \frac{{\partial^2 f}}{{\partial x^2}} = 2 )

( \frac{{\partial^2 f}}{{\partial y^2}} = 2 )

( \frac{{\partial^2 f}}{{\partial x \partial y}} = 0 )

At the critical point ( (-2, 0) ):

( D = \frac{{\partial^2 f}}{{\partial x^2}} \cdot \frac{{\partial^2 f}}{{\partial y^2}} - \left( \frac{{\partial^2 f}}{{\partial x \partial y}} \right)^2 = (2)(2) - (0)^2 = 4 )

Since ( D > 0 ) and ( \frac{{\partial^2 f}}{{\partial x^2}} > 0 ), the critical point ( (-2, 0) ) is a local minimum.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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