1.

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 points of extrema which are opposite in sign are

Answer»

`pi`
(-3,3)
`(-OO,(58)/(14))`
none of these

Solution :`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 negativepoint 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 pointof maxima is POSITIVE point of minima is also positive ltrbgt Hence equation `f(x) =3x6(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) in `(-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).


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