Find minimum of function $f(x) = \frac{\left| x-12\right| }{5}+\frac{\sqrt{x^2+25}}{3}$
I tried compute min using by definition of abs function. I consider two cases:
- when $x > 12$ we have: $f_{1}(x) = \frac{x-12}{5}+\frac{\sqrt{x^2+25}}{3}$ using standard method $f_{1}'(x)=0$ for $x_{0}=-15/4$ but $x_{0}$ is not in $ [12, \infty]$
- when $x \leq 12$ we have $f_{2}(x) = \frac{-x+12}{5}+\frac{\sqrt{x^2+25}}{3}$ similarly $f_{2}'(x) = 0$, and $x = 15/4$
So using observation from this method I computed minimum of $f(15/4) = 56/15$.
Does it be a correct way?