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?