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

Options

A. `pi`
B. `-oo,-3cup(3,(29)/(7))`
C. `-oo,(58)/(14)`
D. none of these

Answer

2

Explanation

`f(x)=x^(3)-3(7-a)x^(2)-3(9-a^(2))x+2`
f(X)=`3x^(2)-6(7-a)x-3(9-a^(2))`
for real root `Dge0`
or `49+a^(2)-14a+9-a^(2)ge0 or ale58/14`
when point of minma is negative point of maxima is also negative
Hence equation f(x) =`3xA^(2)-6(7-a)x-3(9-a^(2))` =0 has both roots negative
sum or roots =`2(7-a)lta or a gt 7 ` which is not possible as form (1) `ale58/14`
When point of maxima is positive point of minima is also positive ltrbgt Hence equation `f(x) =3x^(2)-6(7-a)x-3(9-a^(2))=0` has both roots positive
sum roots =`2(7-a)gt0 or alt7`
Also product of roots is positive or `-(9-a^(2))gt0 or a^(32)gt9 or a in (-oo,-3)cup(3,oo)`
From (1),(2) and (3) ain `(-oo,-3)cup(3,58//14)`
For points of extrema of opposite sign equation (1) has roots of opposite sign
Thus a in (-3,3).