1.

The set of all values of the parameter 'a' for which the points of minimum of the function y=1+a^(2)x-x^(3)satisfy the inequality (x^(2)+x+2)/(x^(2)+5x+6)le0is

Answer»

An EMPTY SET
`(-3sqrt(3),-2sqrt(3))`
`(2sqrt(3),3sqrt(3))`
`(-3sqrt(3),-2sqrt(3))uu(2sqrt(3),3sqrt(3))`

Solution :`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))`


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