Proof that the exponential function is convex

[L Navarro H]

Homework Statement

f(x)=e^(ax)
where a>0

Relevant Equations

A function f(x) is convex if the statement that is into the question marks proofs

I try to proof it but i got stuck right here, i want your opinions
Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?

 

[etotheipi]

Isn't it sufficient to say that $f(x)$ is convex on $(-\infty, \infty)$ if $f''(x) > 0$ for all $x$ in that interval? If $f(x) = e^{ax}$ then $f''(x) = a^2 e^{ax} > 0, \forall x \in \mathbb{R}$.

 

[Mark44]

Relevant Equations:: A function f(x) is convex if the statement that is into the question marks proofs

What does the above mean?
It's hardly an equation, let alone relevant.

 

[pbuk]

Isn't it sufficient to say that $f(x)$ is convex on $(-\infty, \infty)$ if $f''(x) > 0$ for all $x$ in that interval? If $f(x) = e^{ax}$ then $f''(x) = a^2 e^{ax} > 0, \forall x \in \mathbb{R}$.

> gives strictly convex, we don't need that

 

[etotheipi]

> gives strictly convex, we don't need that

Well I suppose that's true , but I did say 'if' and not 'iff'! So what I wrote is true st`atement

 

[Buzz Bloom]

https://en.wikipedia.org/wiki/Convex_function

. . . called convex if the line segment between any two points on the graph of the function lies above the graph between the two points.​