Originally Posted by

**sprinks13** hello,

I have 2 problems here:

1) Let I = [0, pi/2] and let f:I->R be defined as f(x) = sup{x^2, cos x} for x E I. Show that their exists an absolute min point at u E I for f on I. Show that u is the solution to the equation cos x = x^2.

- attempt at a solution:

define f(x) = cos x when cos x>= x^2

= x^2 when x^2 > cos x

let x be such that f(x) = cos x. thus cos x is continuous for all x in R

let x be such that f(x) = x^2. x^2 is a polynomial and is continuous for all x in R

__Does this imply that f(x) is continuous?__

assuming that I proved that f(x) is continuous...

then since I is bounded and f is continuous on I, there exists an absolute min u in I for f on I.

i know that cos x = 1 @ 0 and x^2 = 0 at 0, so then f(0) = 1.

cos x decreases to 0, and x^2 increase to pi^2/4 (both monotonically to x = pi/2).

....

2) suppose f:R->R is continuous on R and that lim(x->infinity)f = 0 and lim(x-> - infinity) f = 0. Prove that f is bounded and that it attains either a minimum or a maximum.

- attempt at a solution:

since I = R is an interval, and f is continuous on I, then f(I) is also an interval.

show that f(I) is bounded.

NO idea

thanks