Prove that $e^\frac{x+y}{2} < \frac{1}{2}(e^x+e^y)$ for $x\neq y$ Should I use the taylor series expansion of the exponential function?

In general if $f''>0$ and $x>y$ then there exist $a\in ([x+y]/2,x)$ and $b\in (y,[x+y]/2)$ and $c\in (a.b)$ such that $$[f(x)+f(y)]/2 -f([x+y]/2)=$$ $$[f(x)-f([x+y]/2)]/2-[f([x+y]/2)-f(y)]/2=$$ $$=[(x-y)f'(a)]/4-[(x-y)f'(b)]/4=(x-y)^2f''(c)/4>0.$$
