TITLE: Finding minimal value of $\sqrt{5x^2 - 40x + 85} + \sqrt{5x^2 - 24x + 53}$ without using derivatives
QUESTION [6 upvotes]: Find minimal value of $f(x) = \sqrt{5x^2 - 40x + 85} + \sqrt{5x^2 - 24x + 53}$.

I can solve it using derivatives. Is there any other way to solve it? For example using some popular inequalities?

REPLY [2 votes]: As suggested by mathlove, if we set $A=\left(\frac{12}{5},\frac{11}{5}\right), B=(x,0)$ and $C=\left(4,1\right)$, we have:
$$ f(x) = \sqrt{5}\left( AB+BC \right) $$
that is minimized when $B$ belongs to the $AC$-line.