Given a concave function $f(x)$ over a convex set $S$, so $f(a)+f(b) \leq 2 \times f (\frac{a+b}{2})$. Here, $a$ and $b$ lies in the same convex set $S$.
I know that the maximum value of $f(a)+f(b)$ is possible when $a=b$. But I don't know how to argue for it. Though I have some intuition, I need formal arguments to justify it.
For eg. the $\sin$ function is concave in the interval $[0, \pi]$, so $\sin(x)+ \sin (y) \leq 2 \times \sin\left(\frac{x+y}{2}\right)$. So, can we say that maximum of $\sin(x)+ \sin (y)$ is possible if $x=y$ ?
Here is a hint.
Notice that the line connecting two points $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$ lies completely below the curve $y=f(x)$. The midpoint of this line is $\left(\frac{x_{1}+x_{2}}{2},\frac{f(x_{1})+f(x_{2})}{2}\right)$