Let f(x) =`x^(3)-3(7-a)X^(2)-3(9-a^(2))x+2`
The values of parameter a if f(x) has a positive point of local maxima are

Explanation

`f'(x) =3 [x^(2) +2(a-7)x+(a^(2)-9)]`
Let `alpha , beta` be roots of f(x) =0 has let `alpha` be the smaller root. Examining sigh change of f(x)

Maxima occurs at smaller root `alpha` which has to be positive . This basically implies that both roots of f(x) =0 must be positive and distinct.
` (i) " "D gt0 " "rArr " "a lt (29)/(7)`
`(ii) " " -(b)/(2a) gt 0 " "rArr " "a lt 7`
`(iii) " " f'(0) gt 0 " "rArr " "a in (-oo , -3)uu (3,oo)`
`"from (i), (ii) and (iii) "" "rArr " " a in (-oo, -3)uu (3, (29)/(7))`