- #1
Calabi
- 140
- 2
Homework Statement
Let be ##f : V \rightarrow \mathbb{R}## a ##C^{1}## function define on a neighbourhood V of the unit sphere ##S = S_{n-1}##(in ##\mathbb{R}^{n}## with its euclidian structure.).
By compacity it exists u in S with ##f(u) = max_{x \in S}f(x) = m##. My goal is to show that ##u## and ##grad(f(u))## are colinear.
Homework Equations
##f(u) = max_{x \in S}f(x)##
The Attempt at a Solution
If m is the maximum in a certain neighbourhood then the gradient is nul so the results is obvious. Then I wroght J the set of all i in ##<1, n>## with ##\frac{\partial f}{\partial i}(u) \neq 0##, if i in J that mean this equality is true on e certain neighbourhood ##V_{i}## of u so I considere the fonction ##x \in V_{i} \rightarrow \frac{x_{i}}{\frac{\partial f}{\partial i}(x)}## and try to show
that they all value a same value on u. But I don't know what to do(perhaps with the implicites function theorem.). and what to do with the nul components. I'm lost.
Could you helpme please?
Thank you in advance and have a nice afternoon.[/B]