# What are the values and types of the critical points, if any, of #f(x, y) = x^3+y^3-3*x*y-7#?

There is a saddle point at

Here, we have

Compute the partial derivatives

The critical point are

Now, compute the second partial derivatives

Therefore,

This point

There is a relative min at

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To find the critical points of ( f(x, y) = x^3 + y^3 - 3xy - 7 ), we first compute the partial derivatives with respect to ( x ) and ( y ), and then set them equal to zero and solve for ( x ) and ( y ). The partial derivatives are:

[ \frac{\partial f}{\partial x} = 3x^2 - 3y ] [ \frac{\partial f}{\partial y} = 3y^2 - 3x ]

Setting these equal to zero and solving for ( x ) and ( y ), we find:

[ 3x^2 - 3y = 0 \implies x^2 = y ] [ 3y^2 - 3x = 0 \implies y^2 = x ]

Solving these simultaneous equations yields critical points at ( (1,1) ) and ( (-1,-1) ). To determine the nature of these critical points, we evaluate the second partial derivatives and use the second derivative test.

The second partial derivatives are:

[ \frac{\partial^2 f}{\partial x^2} = 6x ] [ \frac{\partial^2 f}{\partial y^2} = 6y ] [ \frac{\partial^2 f}{\partial x \partial y} = -3 ]

At the critical point ( (1,1) ), the second partial derivatives are all positive, indicating that ( f(x, y) ) has a local minimum at ( (1,1) ). Similarly, at the critical point ( (-1,-1) ), the second partial derivatives are all positive, indicating a local minimum as well.

Therefore, the values and types of the critical points of ( f(x, y) = x^3 + y^3 - 3xy - 7 ) are:

- ( (1,1) ): Local minimum
- ( (-1,-1) ): 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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