Monotonicity of convex function

In summary, to prove that a convex function has a specific behavior, we must show that for any three points on the function's domain, the ratio of the differences in the function values over the differences in the points is non-decreasing. This can be done with the assumption that the function is convex and a specific condition involving three points. Although the function does not have to be differentiable, we can use the definition of convexity to prove this behavior.
  • #1
hellbike
61
0
[tex]f:(a,\infty)->R[/tex]
i want to prove, that, if function is convex, then:

if exist [tex]x_1 \in R[/tex], exist [tex]x_2>x_1[/tex] : [tex]f(x_2)>f(x_1)[/tex]
then:
for all [tex]x_3>x_2[/tex] for all[tex]x_4>x_3[/tex] : [tex]f(x_4)\ge f(x_3)\ge f(x_2)[/tex]

in other words:
convex function is either decreasing on whole domain, or it starts to increase from some point and then is increasing from that point to the end of domain

We don't assume that function is differential.

With assumption that [tex]f(x_2)>f(x_1)[/tex] i should get something from definition of convex function, but i don't know how to do it.

I'm asking for some tip.
 
Last edited:
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  • #2
First show that convex implies this: If [itex]a < b < c[/itex], then
[tex]
\frac{f(b)-f(a)}{b-a} \le \frac{f(c)-f(b)}{c-b}
[/tex]
 

Related to Monotonicity of convex function

1. What is the definition of monotonicity of a convex function?

The monotonicity of a convex function is a property that describes the behavior of the function as its input increases or decreases. A function is considered monotonic if its output consistently increases or decreases as its input increases.

2. How is monotonicity related to convexity?

Monotonicity is closely related to convexity, as a convex function must also be monotonic. This means that a convex function cannot have any local minima or maxima, and its graph must be continuously increasing or decreasing.

3. What are the implications of monotonicity in convex optimization?

In convex optimization, monotonicity is a desirable property as it guarantees that the objective function will always increase or decrease in the direction of the optimal solution. This simplifies the optimization process and allows for more efficient algorithms.

4. Can a function be convex but not monotonic?

Yes, it is possible for a function to be convex but not monotonic. This means that the function has a convex shape, but it may have local minima or maxima, and its graph may have regions that are neither increasing nor decreasing.

5. How can monotonicity be tested for a convex function?

To test for monotonicity of a convex function, one can look for any changes in the sign of the first derivative of the function. If the first derivative is always positive or always negative, then the function is monotonic. Additionally, the second derivative of a convex function must always be non-negative.

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