Inequalities with function $e^{x^2+e^{x^2}}$

Question

Let $f(x)=e^{x^2+e^{x^2}}$ for $x\in\mathbb{R}$. How to prove that for any $a,b>0$, $a\neq b$ the following inequalites hold

$$(b-a)f(\frac{a+b}{2}) < \int_a^b f(x)\ dx < (b-a)\frac{f(a)+f(b)}{2}$$

I can't connect convexity with integrability, which is obviously needed here. I just don't see how this integral is between $f(tx+(1-t)y)$ and $tf(x)+(1-t)f(y)$.

Answer 1

Hint: Show $f$ is convex and appeal to Hermite-Hadamard Inequality