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Homework Help: Maxima and Minima of a function of several variables

  1. Aug 17, 2013 #1
    1. The problem statement, all variables and given/known data

    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

    2. Relevant equations

    3. 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 dont have the function g(y)?, how can I find g(y)?
  2. jcsd
  3. Aug 17, 2013 #2


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    Staff: Mentor

    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.
  4. Aug 17, 2013 #3
    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?
  5. Aug 17, 2013 #4


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    Staff: Mentor

    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.
  6. Aug 17, 2013 #5
    That`s what my teacher told me, but you are right it doesn´t make sense.
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