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I have a function $f$ defined on $]0,+\infty[\times[0,1] \rightarrow [0,1]$. For the moment let us say that it is smooth enough.

I am looking to find a minimum of this function. What I was told to do (but I do not think it is right, though I cannot find a counter example) is to:

Compute $\frac{\partial f}{\partial x}(x,y) = 0$, this gives me a unique $x_{\min} = h(y)$, then to study $g: y \mapsto f(h(y),y)$, and to see when it is minimum.

My questions:

  • Is it right?
  • If yes, do you know the theorem I should look for?
  • Otherwise, do you have a counter example (even if it means different hypothesis)?
  • If it is true with a function smooth enough, can you tell me the minimal hypothesis needed (and a minimal counter example)?

EDIT: I am especially interested in the minimal hypothesis that make this true, and a minimal counter example when those hypothesis are not matched (questions 3 and 4).

Thanks


Additional informations: there is $a,b>0$, $\frac{\partial f}{\partial y}(x,y) = a - bx$. (meaning the minimum seems to be necessarily for $y=0$ or $y=1$, depending on $x$ so here it works).

I know of the theorem stating that we should look for every point $(x_0,y_0)$ such that $\frac{\partial f}{\partial x}(x_0,y_0) =\frac{\partial f}{\partial y}(x_0,y_0) = 0$, but in this example we never have this condition for the second variable. Do you know of another theorem valid on a compact?

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    I don't understand: if there is *no* critical point at all, there cannot be any *interior* maximum/minimum point. But the point is: are you sure that what you are looking for does exist?2012-06-25
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    @Siminore, The point is that the minimum point will not be on the \emph{interior}, but with $y=0$ or $y=1$. And I do not know if my way is the way to study it formally.2012-06-25

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