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Q.
Value of parameter a for which the function $f(x)=x^3+3(a-7) x^2+3\left(a^2-9\right) x-1$ has a positive point of maximum can be
Application of Derivatives
Solution:
$\Theta f(x)=x^3+3(a-7) x^2+3\left(a^2-9\right) x-1 $
$\Rightarrow f^{\prime}(x)=3 x^2+6(a-7) x+3\left(a^2-9\right)$
For $f(x)$ to have a positive point of maximum, both roots of $f^{\prime}(x)$ should be distinct and greater than 0 $\therefore$ Both roots of $x^2+2(a-7) x+\left(a^2-9\right)=0$ are greater than zero
(i)$D>0 \Rightarrow 4(a-7)^2-4\left(a^2-9\right)>0 $
$ \Rightarrow -14 a+58>0 \Rightarrow a<\frac{29}{7}$
(ii) $a^2-9>0 \Rightarrow a<-3$ or $a>3$
(iii) $\frac{-2(a-7)}{2 \cdot 1}>0 \Rightarrow a<7 $
$\therefore a \in(-\infty,-3) \cup\left(3, \frac{29}{4}\right) $