Maximum and Minimum Question?

I'm frustrated beyond belief with a proof.

Suppose we have an continuous even function with a domain of all real numbers. Now this function has limit as x goes to negative infinty equal to l and the limit as x goes to positive infinty is also equal to l.

I want to show that this function will either have a maximum or a minimum.

I'm not sure at all how to show this rigorously since I don't know how to apply the definition of a limit to limits at infinity. I think it has to do with bounds. And I need to do this without first derivative test.
 Recognitions: Gold Member Staff Emeritus Aren't the specifications you give consistent with the constant function f(x) = 1?
 Did you think my l was a 1. Maybe I should write it differently. lim (x-> -oo) f(x) = lim (x-> oo) f(x) = a and a is even. Want to show f has either a minimum or a maximum.

Maximum and Minimum Question?

intuitively this makes sense, but rigorously you could show that unless it is a consant function (for example y = 1) then there must be a point where it switches between a positive and negative slope. Im not entirely sure what level of "rigorousness" you want.
 Recognitions: Gold Member Science Advisor Staff Emeritus First, you still have the problem that was pointed out by both selfadjoint and T@p: The constant function f(x)= a satisfies your conditions but does not have a maximum or minimum so the "theorem" as you stated it is not true. If f(x) is NOT A CONSTANT FUNCTION, then there exist some x0 such that f(x0) is not equal to a and so is either larger than or less than a. Assume f(x0)> a. Since limit as x-> infinity f(x)= a, there exist some x1> x0 such that f(x1)< f(x0). Similarly, since limit as x-> -infinity f(x)= a, there exist some x2< x0 such that f(x2)< f(x0). Since f is continuous on the closed and bounded interval [x2, x1] it must have both maximum and minimum values there. Now show that f has a maximum on -infinity to infinity.
 I need help. Find the minimum of y = Absolute value of (sinx + cosx + tanx + cotx + secx + cscx) Thanks Ruth Jackson the_perfect_mom@hotmail.com

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