Question

Calc III - Max/Min question

I'm understanding everything here except for the maximum and minimum part which I denoted with an arrow. Image attached.

Answer 1

Looks like
\[f_y(x,y)=2y+x^2=0\]
\[-2y=x^2\]
\[y=\frac{-x^2}{2}\]
Is a whole curve of negative y values.

Answer 2

That wasn't really the question, what I was stuck on is how they are finding the max and min in step 2 of #31

Answer 3

Could you write what you think the phrase "minimum when y= 1/2" is referring to? What is the situation?

Answer 4

Well, for the problem I'm trying to find the absolute maximum and minimum values of f on the set D.

Answer 5

Right, so you have x' inflections where x=0 or y=-1 and y' inflections on the curve y=-x^2/2. It is correct that the only inflection point where they intersect is at (0,0) and that is the absolute minimum because f(-1,1),f(-1,-1),f(1,-1),f(1,1) are all > 4. The question then is where are the maximums?

Answer 6

Basically , what you need to test is the end points and the points at which the derivative tests denote a max or min.

Answer 7

so i assume you understand the tests?

Answer 8

The only candidates for the maximums are on the boundary of D.

Answer 9

thats what i mean the boundaries end points can be max and minimums

Answer 10

For instance, in part 2, where x=-1, f(-1,y)=y^2+y+5
Where do i go from there?

Answer 11

That curve starts at 5 when y is -1, goes down to 4.74 where y=-1/2 and up to 7 at y = 1.
The same is true for the curve where x=+1

Answer 12

*4.75

Answer 13

The mistake on the paper says "minimum when y=1/2" when it should read "minimum when y=-1/2.

Answer 14

Sorry if this seems stupid, but how can the curve start at 5 when the region is bounded at 1?

Answer 15

The way you know this is a local min is from the expression
\[f(\pm1,y)=y^2+y+5\]
\[f'(\pm1,y)=2y+1; \text{set equal to 0};y=\frac{-1}{2}\]

Answer 16

f(-1,-1)=5

Answer 17

The Domain of x and y are |x|<=1;|y|<=1. The range of the function is from 4 to 7.

Answer 18

Oh, I see now, I didn't know that I was supposed to take another derivative of
f′(±1,y)

Answer 19

Thank you for your help!

Answer 20

Cheers