How to proove that that e^x is convex

In summary, the conversation revolves around determining if f(x)=e^x is a convex function and providing a proof for it. The attempt at a solution involves using the definition of convexity and substituting values to show that the function satisfies the definition. The conversation also mentions the possibility of using the second derivative to prove convexity.
  • #1
Rabolisk
6
0

Homework Statement



I have to determine if [e][/x] is a convex function. If it is then show proof. I know its a convex function by looking at the graph, Iam stuck at prooving it mathematically though.


Homework Equations



The function is f(x)=e^x.


The Attempt at a Solution


I am certain that the function is convex. I'am having trouble proving it though.

Assuming we pick a point and call it x0, then a lies to the left of x0 and point b lies to the right.

f(ta+(1-t)b)<=t*f(a)+(1-t)*f(b)

Once we substitute we get.
e^((ta+(1-t)b))<=t*e^a+(1-t)*e^b

I'am stuck at proving how this inequality is true.

Thanks!
 
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  • #2
If you are required to use the definition of "convex" then use the fact that
[tex]e^{ta+ (1-t)b}= e^{ta}e^be^{-bt}[/tex]

Or are you allowed to use the fact that a function is convex if and only if its second derivative is positive for all x?
 

FAQ: How to proove that that e^x is convex

What is the definition of a convex function?

A convex function is a function where the line segment between any two points on the graph of the function lies above or on the graph, meaning the function is never concave or curved inward.

How do you prove that e^x is convex?

To prove that e^x is convex, we can use the definition of convexity. We can show that the line segment between any two points on the graph of e^x lies above or on the graph by using the second derivative test. If the second derivative of e^x is always positive, then the function is convex.

3. What is the second derivative of e^x?

The second derivative of e^x is equal to e^x. This can be found by taking the derivative of the first derivative of e^x, which is also e^x. Since e^x is always positive, we can conclude that the second derivative of e^x is always positive, making the function convex.

4. Can you use other methods to prove that e^x is convex?

Yes, there are other methods to prove that e^x is convex. One method is to show that the function is continuous and has a positive second derivative at every point. Another method is to use the convexity criteria, which states that if the second derivative of a function is always positive, then the function is convex.

5. Why is it important to prove that e^x is convex?

Proving that e^x is convex is important because it is a fundamental property of the exponential function. It also has many practical applications in fields such as economics, physics, and statistics. Additionally, understanding convexity can help us analyze and optimize functions in various contexts.

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