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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 |
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Answer» `pi` 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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