Independent of original function I came across the following problem in a novel I was reading (it was poorly translated) so I decided to see if it was a valid problem.
The graph of a differentiable function $y=f(x)$ is concaved downward over the interval $[a,b]$. Let $P=(x_0,f(x_0))$ with $a\leq x_0 \leq b$. If the area bounded by $f(x)$, the tangent at $P$, $x=a$ and $x=b$, is minimized. Prove that the x-coordinate of the point $P$ is independent of the function $f(x)$.
I gotten to a point where I must minimize $f'(x_0)(\frac{a+b}{2}-x_0)+f(x_0)$. It seems that based on the question, we must have $x_0=\frac{a+b}{2}$. However, I can't seem to figure out why this is true.

You get
$$
\frac{d}{dx_0}\left(f'(x_0)\left(\frac{a+b}{2}-x_0\right)+f(x_0)\right)=f''(x_0)\left(\frac{a+b}{2}-x_0\right)
$$
