The values of parameter `a` for which the point of minimum of the function `f(x)=1+a^2x-x^3` satisfies the inequality `(x^2+x+2)/(x^2+5x+6)<0a r e` `(2sqrt(3),3sqrt(3))` (b) `-3sqrt(3),-2sqrt(3))` `(-2sqrt(3),3sqrt(3))` (d) `(-2sqrt(2),2sqrt(3))`

Options

A. An empty set
B. `(-3sqrt(3),-2sqrt(3))`
C. `(2sqrt(3),3sqrt(3))`
D. `(-3sqrt(3),-2sqrt(3))uu(2sqrt(3),3sqrt(3))`

Answer

C

Explanation

`y=1+a^(2)x-x^(3)implies(dy)/(dx)=a^(2)-3x^(2),(d^(2)y)/(dx^(2))=-6x" "therefore(dy)/(dx)=0impliesx=pm(a)/(sqrt(3))`
Also `(x^(2)+x+2)/(x^(2)+5x+6)le0impliesx^(2)+5x+6lt0impliesx in (-3,-2)`
If `a gt 0` than `x=-(a)/(sqrt(3))` is point of minima `implies - (a)/(sqrt(3)) in (-3,-2)implies a in (2sqrt(3),3sqrt(3))`
If `a lt 0` then `x=(a)/(sqrt(3))` is point of minima `implies(a)/(sqrt(3))in (-3,-2)implies a in (-3sqrt(3),-2sqrt(3))`