Maxima and Minima of a function of several variables

[Sly37]

Homework Statement

Find the maxima and minima of:

f(x,y)=(1/2)*x^2 + g(y)

g∈⊂ (δ⊂ ℝ )

in this region
Ω={(x,y)∈ℝ2 / (1/2)*x^2 + y^2 ≤ 1 }

hint: g: δ⊆ ℝ→ℝ

The absolute min of f in Ω is 0
The absolute max of f in Ω is 1

Homework Equations

The Attempt at a Solution

I have the parameterization of the region: √2 *cosθ and senθ
I also know ∇f(x,y)=(x,g`(y))=(0,0)
x=0, g´(y)=0
Hessian Matrix= [ 1 0
0 g``(y) ]
Determinat of the hessian matrix=g``(y)

How can I complete this problem if I don't have the function g(y)?, how can I find g(y)?

 

[mfb]

g∈⊂ (δ⊂ ℝ )

What does that mean?

The new parameter set is for the border of Ω only? For maxima/minima inside, I would keep x and y. It is easy to find a condition for x there.

How can I complete this problem if I don't have the function g(y)?, how can I find g(y)?

That is strange, indeed.

 

[Sly37]

What does that mean?

The new parameter set is for the border of Ω only? For maxima/minima inside, I would keep x and y. It is easy to find a condition for x there.

That is strange, indeed.

I believe that means that g belongs to a subset of ℝ.

Yes, the new parameter is just for the border.

All the possible critical points would have to be (0,?). and ? being the result gotten from g`(y)=0, right?

 

[mfb]

I don't recognize "belongs to" as a mathematical term. g is certainly not a subset of ℝ, as it is a function ℝ→ℝ.

All the possible critical points would have to be (0,?). and ? being the result gotten from g`(y)=0, right?

Right, assuming g is differentiable.

 

[Sly37]

I don't recognize "belongs to" as a mathematical term. g is certainly not a subset of ℝ, as it is a function ℝ→ℝ.

Right, assuming g is differentiable.

That`s what my teacher told me, but you are right it doesn´t make sense.