Proof of Unit Sphere Homework Statement

[MIT2014]

Homework Statement

f is a polynomial with n variables (x₁, x₂, ... , x_n) with real coefficients. Let S^n-1 = {x E Rⁿ | x₁² + x₂² + ... + x_n² = 1} (n-1 unit sphere). Show that $$\exists$$ b,c E S^n-1 such that m = f(b) $$\leq$$ f(x) $$\leq$$ f(c) $$\leq$$ = M for all x E S^n-1.

If f(x₁, ... , x_n) = a₁x₁ + a₂x₂ + ... + a_nx_n with (a₁ ,..., a_n) constants, determine m and M.

If n$$\geq$$2, show that $$\exists$$ y E S^n-1 such that f(y) = f(-y)

Homework Equations

The Attempt at a Solution

 

[micromass]

So what did you already try??

 

[MIT2014]

Honestly, I have no clue about this one.

 

[micromass]

Hint: the sphere is compact.

 

[MIT2014]

Okay, here goes my attempt at this:
part 1:
S^n-1 is both complete and bounded (can we assume these two things, or do we have to prove them). Thus, S^n-1 must be a compact. Since f is a real function on compact S^n-1 into R^k, f is bounded. Thus, it follows that there exist b,c E S^n-1 such that f(b) $$\leq$$ f(x) $$\leq$$ f(c)

part 2:
Let a_i = max(a₁, ... , a_n). Then let x_i = 1 and all other x_n's = 0. Then a_ix_i = M.

On the other hand, if we let a_i = max(a₁, ... , a_n), but this time let x_i=-1, then a_ix_i = m.

part 3:
i need help with this part

 

[micromass]

For number 2, I don't really see how you got that? Who says the maximum is reached at (1,0,0,...,0)??

For number 3: think intermediate value theorem.