The function $f (x) = | ax − b | + c | x | ∀ x ∈ (−∞, ∞)$, where $a > 0, b > 0, c > 0$, assumes its minimum value only at one point if

$a ≠ c$

$f(x)=\left\{\begin{matrix}b-(a+c)x&,&x<0\\b+(c-a)x&,&0≤x<\frac{b}{a}\\(a+c)x+b&,&x≥\frac{b}{a}\end{matrix}\right.$

These figures clearly indicate that for exactly one point of minima, $a ≠ c$.