The function $$f\left( x \right) = \left| {px - q} \right| + r\left| x \right|,x \in \left( { - \infty ,\infty } \right)$$       where $$p > 0,\,q > 0,\,r > 0$$     assumes its minimum value only on one point if

Solution :
$$f(x) = \left| {px - q} \right| + r\left| x \right|$$
\[ = \left\{ {\begin{array}{*{20}{c}} { - px + q - rx,}\\ { - px + q + rx,}\\ {px - q + rx,} \end{array}\,\,\begin{array}{*{20}{c}} {x \le 0}\\ {0 < x \le \frac{q}{p}}\\ {\frac{q}{p} < x} \end{array}} \right.\]
$$\eqalign{ & {\text{For}}\,r = p,f'\left( x \right) < 0\,{\text{if}}\,x < 0 \cr & = 0\,{\text{if}}\,0 < x < \frac{q}{p} \cr & > 0\,{\text{if}}\,x > \frac{q}{p} \cr} $$

From graph (i) infinite many points for min value of $$f\left( x \right)$$
$$\eqalign{ & {\text{for}}\,r < p,f'\left( x \right) < 0\,{\text{if}}\,x \leqslant 0 \cr & < 0\,{\text{if}}\,{\text{0 < }}\,x \leqslant \frac{q}{p} \cr & > 0\,{\text{if}}\,x < \frac{q}{p} \cr} $$

From graph (ii) only pt. of min of $$f\left( x \right)$$ at $$x = \frac{q}{p}$$
$$\eqalign{ & {\text{For}}\,r > p,f'\left( x \right) < 0\,{\text{if}}\,x \leqslant 0 \cr & > 0\,{\text{if}}\,0 < x \cr} $$

From graph (iii) only one pt. of min of $$f\left( x \right)$$  at $$x = 0$$