Proving Convexity of Bounded Function F

In summary, according to the author, the function F is monotonic increasing and has a constant value at a point y<"a"<x.
  • #1
talolard
125
0

Homework Statement


Hey, the original question is not in english, so I am translating. So just to make sure I'm understood, i take convex to mean that the graph of the function is below the tangent.

The question:
Let F be a convex function and F is bounded from above by some number C, prove that F is static (again, my translation, by static I mean that for every x F(X)=a)



The Attempt at a Solution



I don't think I'm close, but I am stumped, some mild hin tto point me in the right direction would be great.
we will write F as a taylor expansion:
[tex] f(x_0)+f'(x_0)(x-x_0) > f(x_0)+f'(x_0)(x-x_0) + \frac {f''(c)(x-x_0)^2}{2} <a \iff [/tex]
[tex]
0 > \frac {f''(c)(x-x_0)^2}{2} <a - f(x_0)+f'(x_0)(x-x_0) \iff [/tex]
[tex]
0 >\frac {f''(c)}{2} < \frac {a}{(x-x_0)^2} - \frac {f(x_0)+f'(x_0)}{x-x_0} [/tex]





 
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  • #2
I don't think Taylor's theorem is required here. The other flaw with your approach is that you are assuming some value for a beforehand in trying to show the function is constant. I think a better approach is to show that f(x) = f(y) for every real x,y.

I think it's a lot easier if you consider two points x and y and suppose x < y. If f(y) > f(x), what can you say about the behavior of f as x approaches infinity? Similarly, what if f(y) < f(x)? Remember in working with these two cases, you are trying to obtain a contradiction if your goal is to show the function is constant. But there is really only one statement in the hypothesis you can directly contradict.
 
  • #3
The definition I learned of convex is that the graph lies above the tangent. I don't think the problem works with your definition: for example, f(x)=-x2 is bounded above by 0, and lies below its tangent.
 
  • #4
The function doesn't need to be strictly convex.
 
  • #5
snipez90 said:
I don't think Taylor's theorem is required here. The other flaw with your approach is that you are assuming some value for a beforehand in trying to show the function is constant. I think a better approach is to show that f(x) = f(y) for every real x,y.

I think it's a lot easier if you consider two points x and y and suppose x < y. If f(y) > f(x), what can you say about the behavior of f as x approaches infinity? Similarly, what if f(y) < f(x)? Remember in working with these two cases, you are trying to obtain a contradiction if your goal is to show the function is constant. But there is really only one statement in the hypothesis you can directly contradict.

Ok, it took me a few days to work this over and I'm still not there. Here's what I have so far:

Since F is convex then every point has one sided deriviatives such that
[tex] x>y f_{-}'(y) \leq f_{+}'(y) \leq f_{-}'(x) \leq f_{+}'(x) [/tex]
which means that the function is monotonic increasing.
let x>y then [tex] lim_{x-> \infty} f(x) = c [/tex] because the function is monotonic increasing and bounded from above. But this means that
[tex] lim_{x-> \infty} \frac {f(x) -f(y)}{x-y}= lim_{x-> \infty} \frac {c -f(y)}{x-y} =0 [/tex] Which means that there exists a point y<"a"<x where [tex] f'(a)=0 [/tex] but this means that for any x<a [tex] f'(x)=0 [/tex]

Now take y>x>a. Then [tex] lim_{x-> \infty} f(x) = c and f(y)=c [/tex] so for all x after a certain point [tex] f'(x)=0 [/tex] then all points before that point must also have [tex] f'(x)=0 [/tex] because of the monotonity of f. Then f is constant.

Is that correct?
Thanks
 

FAQ: Proving Convexity of Bounded Function F

What is a bounded function?

A bounded function is a function that has both an upper and lower limit. This means that the function values cannot exceed a certain value, known as the upper bound, and cannot be less than a certain value, known as the lower bound.

How do you define convexity?

Convexity is a mathematical property of a function that describes its curvature. A function is convex if every line segment connecting two points on the function lies above or on the graph of the function.

How can I prove the convexity of a bounded function?

To prove the convexity of a bounded function, you can use the definition of convexity and show that for any two points on the function, the line segment connecting them lies above or on the function. You can also use the second derivative test to check for convexity.

What is the second derivative test?

The second derivative test is a method used to determine the concavity and convexity of a function at a given point. It involves taking the second derivative of the function and evaluating it at the point of interest. If the second derivative is positive, the function is convex, and if it is negative, the function is concave at that point.

Can a function be both bounded and non-convex?

Yes, a function can be both bounded and non-convex. Boundedness only refers to the upper and lower limits of the function, while convexity refers to the shape of the function. A function can have a bounded range but still have a concave or non-convex shape.

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