Find the local maximum and minimum values and saddle point(s) of the function. if you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (enter your answers as a ,-separated list. if an answer does not exist, enter dne.) f(x, y) = x3 + y3 − 3x2 − 3y2 − 9x

A(-1,0) is a local maximum point.

B(-1,2) is saddle point.

C(3,0) is also a saddle point

D(3,2) is the minimum point.

Step-by-step explanation:

Given information:

The function :

The first partial derivative with respect to x is:

And the partial derivative with respect to y is:

Now equate the above derivatives to zero, to obtain the system of equation:

On solving the two equations:

The critical points are : A(-1,0), B(-1,-2), C(3,0) and D(3,2)

Now calculate the discriminant

Here,

On putting the above values we get:

At, A(-1,0)

Hence , A(-1,0) is a local maximum point.

At, B(-1,2)

Hence B(-1,2) is saddle point.

At, C(3,0)

Hence C(3,0) is also a saddle point

At, D(3,2)

Hence , D(3,2) is the minimum point.

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A(-1,0) is a local maximum point.

B(-1,0)  is a saddle point

C(3,0)  is a saddle point

D(3,2) is a local minimum point.

Step-by-step explanation:

The given function is  

The first partial derivative with respect to x is  

The first partial derivative with respect to y is  

We now set each equation to zero to obtain the system of equations;

Solving the two equations simultaneously, gives;

 and

The critical points are

A(-1,0), B(-1,2),C(3,0),and D(3,2).

Now, we need to calculate the discriminant,

But, we would have to calculate the second partial derivatives first.

At A(-1,0),

and

Hence A(-1,0) is a local maximum point.

See graph

At B(-1,2);

Hence, B(-1,0) is neither a local maximum or a local minimum point.

This is a saddle point.

At C(3,0)

Hence, C(3,0) is neither a local minimum or maximum point. It is a saddle point.

At D(3,2),

and

Hence D(3,2) is a local minimum point.

See graph in attachment.