F(x, y)=x^2+y^2-6x-6y+41. how many critical points does g have in r22. if there is a local minimum, what is the value of the discriminant d at that point? 3. if there is a local maximum, what is the value of the discriminant d at that point? 4. if there is a saddle point, what is the value of the discriminant d at that point? 5. what is the maximum value of f in r2? 6. what is the minimum value of f in r2?

1) There is only a critical point at (3,3).

2) If there is a local minimum, the value of the discriminant must be D>0

3) If there is a local maximum, the value of the discriminant must be D>0

4) If there is a saddle point, the value of the discriminant must be D<0

5) There is not a local maximum of f

6) There is a local minimum at (3,3). f(3,3)=23

Step-by-step explanation:

We have the fuction:

Its partial derivatives are:

When

0=2x-6 ⇒ 2x=6 ⇒x=3

When

0=2y-6 ⇒ 2y=6 ⇒y=3

The critical point is (3,3)

Its second order derivatives are:

The value of the discriminant is

=2×2-0=4

As D>0 and , there is a local minimun at (3,3)

The value of f(x,y) at (3,3) is:

351°f

step-by-step explanation:

the answer is 40.

proof: