How to proove that that e^x is convex

[Rabolisk]

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!

 

[HallsofIvy]

If you are required to use the definition of "convex" then use the fact that
$$e^{ta+ (1-t)b}= e^{ta}e^be^{-bt}$$

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?