The function $f(x) = |px - q| + r|x|,\ x \in ( - \infty,\ \infty)$, where $p 0,\ q 0,\ r 0$ assume its minimum value only at one point if

Question

The function $f(x) = |px - q| + r|x|,\ x \in ( - \infty,\ \infty)$, where $p > 0,\ q > 0,\ r > 0$ assume its minimum value only at one point if

Dr. Abubacker · Accepted Answer

(c)
We have, $f(x) = |px - q| + r|x|,\ x \in ( - \infty,\ \infty)$
$= \left\{ \begin{array}{r}
 - px + q - rx,\ \ x \leq 0 \
 - px + q + rx,\ \ \ \ \ 0 < x < q/p \
px - q + rx,\ \ \ q/p < x
\end{array} ight.\$
[image:6_image9.emf]
Thus, $f$ has infinite points if minimum, if $r = p$
In case, $p 
eq r$, then $x = 0$ is point of minimum, if $r > p$ and $x = \frac{q}{p}$ is point of minimum, if $r < p$

Q. The function $f(x) = |px - q| + r|x|,\ x \in ( - \infty,\ \infty)$, where $p > 0,\ q > 0,\ r > 0$ assume its minimum value only at one point if

Explanation

Reviewed by Dr. Abubacker

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