# Thread: Prove that f is strictly increasing

1. ## Prove that f is strictly increasing

f:[0,∞) → ℝ, f(x) = e^(x^2)

In order to prove that f is strictly increasing, do I have to show that f'(x) is always positive?

I think f'(x)=[(x^2)e^(x^2)]/e + 2xe^(x^2)lne
I could have gotten that horribly wrong, so correct me if I have.

Then how to prove that this is strictly positive?

Thanks.

2. Originally Posted by feyomi
f:[0,∞) → ℝ, f(x) = e^(x^2)

In order to prove that f is strictly increasing, do I have to show that f'(x) is always positive?

I think f'(x)=[(x^2)e^(x^2)]/e + 2xe^(x^2)lne
I could have gotten that horribly wrong, so correct me if I have.

Then how to prove that this is strictly positive?

Thanks.

Sorry just to add to that. Is it enough to say that each term in f'(x) is positive since xϵ[0,∞) and since e>0 and so f'(x)>0, thus f is strictly increasing?

Thanks

3. Originally Posted by feyomi
f:[0,∞) → ℝ, f(x) = e^(x^2)

In order to prove that f is strictly increasing, do I have to show that f'(x) is always positive?

I think f'(x)=[(x^2)e^(x^2)]/e + 2xe^(x^2)lne
I could have gotten that horribly wrong, so correct me if I have.

Then how to prove that this is strictly positive?

Thanks.
$f(x) = e^{x^2}$

$f'(x) = 2x e^{x^2}$

for $x > 0$ , $f'(x) > 0$

4. Hahaha of course. Silly me.
Thank you.