The minimum value of the function,
`f(x)=x^(3//2)+x^(-3//2)-4(x+(1)/(x)).` For all permissible real values of x is

Options

A. `-10`
B. `-6`
C. `-7`
D. `-8`

Answer

A

Explanation

We have
`f(x)=x^(3//2)+x^(-3//2)-4(x+1/x)`
`rArr f(x)=(sqrt(x)+(1)/(sqrt(x)))^3-3(sqrt(x)+(1)/(sqrt(x)))-4{(sqrt(x)+(1)/(sqrt(x)))^2-2}`
`rArr f(x)=(sqrt(x)+(1)/(sqrt(x)))^3-4(sqrt(x)+(1)/(sqrt(x)))^2-3{(sqrt(x)+(1)/(sqrt(x)))+8}`
Clearly ,f(x) is defined for all `x lt 0`
Let `sqrt(x)+(1)/(sqrt(x))=t` Then `t le 2 "for all " x gt 0 `
Also let `g(t)=t^3-4t^2-3t +8` Then
`g(t)=3t^2-8t-3 and g (t)=6t-8`
`rArr g(t) =(3t +1)(t-3) and g''(t)=6t-8`
Clearly g(t)=0 for t=3 and g'' (t) `=10 gt 0`
Thus, g is minimum when t=3 and the minimum value of g is g(3)= -10