Question

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?

Answer 1

Answer:

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:

$f(x,y)=x^2+y^2-6x-6y+41$

Its partial derivatives are:

$f_x=2x-6$

$f_y=2y-6$

When $f_x=0$

0=2x-6 ⇒ 2x=6 ⇒x=3

When $f_y=0$

0=2y-6 ⇒ 2y=6 ⇒y=3

The critical point is (3,3)

Its second order derivatives are:

$f_{xx}=2$

$f_{yy}=2$

$f_{xy}=0$

The value of the discriminant is

$D=f_{xx} \times f_{yy}-(f_{xy})^2$=2×2-0=4

As D>0 and $f_{xx}>0$, there is a local minimun at (3,3)

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

$f(x,y)=x^2+y^2-6x-6y+41=3^2+3^2-6\times3-6\times3+41=9+9-18-18+41=23$