Maximum and Minimum Limits at Infinity Proof?

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SUMMARY

The discussion centers on proving that a continuous even function with limits at both negative and positive infinity equal to a constant \( l \) must have either a maximum or a minimum, provided it is not a constant function. The participants clarify that while the constant function \( f(x) = a \) meets the limit conditions, it does not possess a maximum or minimum. They establish that if \( f(x) \) is not constant, there exist points \( x_0 \), \( x_1 \), and \( x_2 \) such that \( f(x_0) > a \) and \( f(x_1) < f(x_0) \) for \( x_1 > x_0 \), and similarly for \( x_2 < x_0 \). This leads to the conclusion that the function must achieve both maximum and minimum values within the interval defined by \( [x_2, x_1] \).

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  • Understanding of limits at infinity in calculus
  • Knowledge of continuous functions and their properties
  • Familiarity with the concept of even functions
  • Basic understanding of the Intermediate Value Theorem
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Nebula
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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.
 
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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.
 
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. I am not entirely sure what level of "rigorousness" you want.
 
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
 
HallsofIvy said:
Assume f(x0)> a. Since limit as x-> infinity f(x)= a, there exist some x1> x0 such that f(x1)< f(x0).

I think you need to add something about x1 here. Choose it in such a way that if x>x1 then f(x)<f(x0). Similar change with how you select x2.
 

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