Question

If $p,q$and $r$ are any real numbers, then

• A. $max(p,q)<max(p,q,r)$
• B. $min(p,q)=\left(\frac{1}{2}\right)[p+q–|p–q\left|\right)$
• C. $max(p,q)<min(p,q,r)$
• D. none of these

Answer 1

The correct option is B

$min(p,q)=\left(\frac{1}{2}\right)[p+q–|p–q\left|\right)$

Explanation for the correct option

If $p,q,r$ are any real numbers then

$$\begin{gathered} \Rightarrow max(p,q)=\left\{p,p>qandq,q>p\right\} \\ \Rightarrow max(p,q,r)=\left\{p,p>q,q,q>r,r,r>p\right\} \\ \Rightarrow max(p,q)<max(p,q,r)isfalse \\ \Rightarrow |p–q|=\left\{p–q,p\ge qandq–p,p<q\right\} \\ \Rightarrow \left(\frac{1}{2}\right)(p+q–|p–q\left|\right)=\left\{\left(\frac{1}{2}\right)(p+q–p+q),p\ge qand\left(\frac{1}{2}\right)(p+q+p–q),p<q\right\} \\ =\left\{q,p\ge qandp,p<q\right\} \\ \Rightarrow \left(\frac{1}{2}\right)(p+q–|p–q\left|\right)=min(p,q) \end{gathered}$$

Hence option ‘B’ is the correct answer.