I can't prove the next inequality:
Let a,b be in R and b>a. Prove that:
e^b - e^a >= (b-a)*(a+b+2)/2
You have $\displaystyle e^b-e^a=e^{\frac{a+b}{2}}2\sinh\frac{b-a}{2}$. Furthermore, $\displaystyle e^x\geq x+1$ and $\displaystyle \sinh x\geq x$ (since $\displaystyle \sinh x=\int_0^x \cosh t dt\geq \int_0^x 1 dt=x$). Put these together, and you'll get your inequality.