`P=x^3-1/x^3, Q=x-1/x` `x in (1,oo)` then minimum value of `P/(sqrt(3)Q^2)`

Options

A. `2sqrt(2)`
B. `-2sqrt(3)`
C. non-existent
D. non of these

Answer

A

Explanation

We have `p=x^3-1/x^3and Q=x-1/x`
`p=(x-1/x)^3+3(x-1/x)`
`P=Q^3+3Q rArr p/Q^2=Q+3/Q`
Let `Z=P/Q^2=Q +3/2`
Let `Z=P/Q^2+Q+3/Q` Then
`dz/dx=(dQ)/dx-3/Q^2(dQ)/(dx)`
For maximum /minimum we must have
`(dZ)/(dx)=0`
`rArr (dQ)/(dx){1-3/Q^2}=0 rArr Q^2=3 " "[ because (dQ)/(dx)=1+1/x^2 ne 0 ]`
`rArr Q=pm sqrt(3)`
Again `(dz)/(dx)` for different values of Q are shown if Fig. 25

Clearly ,Z is minimum at `Q=sqrt(3)`
The minimum value of Z is given by
`Z=(p)/(Q^2)=(3 sqrt(3)+3 sqrt(3))/(3)=2sqrt(3)`