# How do you find the critical points for #f(x, y) = x^2 + 4x + y^2# and the local max and min?

The point

Evaluate the partial derivatives of the first order:

so the critical points are the solutions of the equations:

so we have that:

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To find the critical points of ( f(x, y) = x^2 + 4x + y^2 ), we need to compute the partial derivatives with respect to ( x ) and ( y ) and then solve the system of equations where both partial derivatives are equal to zero.

Partial derivatives: [ \frac{\partial f}{\partial x} = 2x + 4 ] [ \frac{\partial f}{\partial y} = 2y ]

Setting both partial derivatives equal to zero: [ 2x + 4 = 0 ] [ 2y = 0 ]

Solving these equations gives us: [ x = -2 ] [ y = 0 ]

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

To determine whether this critical point is a local maximum, minimum, or saddle point, we use the second partial derivative test. We compute the second partial derivatives and evaluate them at the critical point.

Second partial derivatives: [ \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 ]

The determinant of the Hessian matrix at ( (-2, 0) ) is ( (2)(2) - (0)^2 = 4 ), which is positive. Also, the second partial derivative with respect to ( x ) is positive at this point.

Since the determinant is positive and the second partial derivative with respect to ( x ) is positive, the critical point ( (-2, 0) ) corresponds to a local minimum for the function ( f(x, y) = x^2 + 4x + y^2 ).

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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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