- #1
MIT2014
- 10
- 0
Homework Statement
f is a polynomial with n variables (x1, x2, ... , xn) with real coefficients. Let Sn-1 = {x E Rn | x12 + x22 + ... + xn2 = 1} (n-1 unit sphere). Show that [tex]\exists[/tex] b,c E Sn-1 such that m = f(b) [tex]\leq[/tex] f(x) [tex]\leq[/tex] f(c) [tex]\leq[/tex] = M for all x E Sn-1.
If f(x1, ... , xn) = a1x1 + a2x2 + ... + anxn with (a1 ,..., an) constants, determine m and M.
If n[tex]\geq[/tex]2, show that [tex]\exists[/tex] y E Sn-1 such that f(y) = f(-y)