Find minimum value of `px+qy` where `p>0, q>0, x>0, y>0` when `xy=r,^2` without using derivatives.

Options

A. `2rsqrt(pq)`
B. `2 pq sqrt(r)`
C. `-2rsqrt(pq)`
D. 2 rpq

Answer

A

Explanation

Given that `xy=r^(2)`
`rarr y = (r^(2))/(x)`
`Let S= px + qy =px +(qr^(2))/(x)`
`rarr (ds)/(dx)=p-(qr^(2))/(x^(2))`
`(ds)/(dx)=0` for maximum or minimum
So , 0= `p-(qr^(2))/(x^(2))`
Now `(d^(2)s)/(dx^(2))=(2qr^(2))/(x^(3))`
Hence S is minimum at x `x=sqrt(q)/(p) r`
`rarr y= (r^(2))/sqrt(q)/(p)r =sqrt(p)/(p)r`
Minimum value of px + qy =`p.sqrt(q)/(p).r + qsqrt(p)/(q).r`
`=sqrt(pqr)+sqrt(pqr)=2rsqrt(pq)`