Question

The values of a for which $f(x)=x^3+3(a-7) x^2+3\left(a^2-9\right) x-1$, have a positive point of maxima is,

• A. $(-\infty,-3)$
• B. $\left(3, \frac{29}{7}\right)$
• C. $(-\infty,-3) \cup(3, \infty)$
• D. $(-\infty,-3) \cup\left(3, \frac{29}{7}\right)$

Answer 1

Answer: (D)

Here $f^{\prime}(x)=3 x^2+6(a-7) x+3\left(a^2-9\right)=0$
$\Rightarrow x=7-a \pm \sqrt{58-14 a}, x_1=7-a+\sqrt{58-14 a}, x_2=7-a-\sqrt{58-14 a}$
$58-14 a>0 \Rightarrow 14 a<58 \Rightarrow a<\frac{29}{7}$
$f^{\prime \prime}(x)=6 x+6(a-7), f^{\prime \prime}\left(x_2\right)<0 $
$\Rightarrow a t x=a_2, f(x)$ has a maxima
$\therefore x_2>0 \Rightarrow 7-a-\sqrt{58-14 a}>0, \sqrt{58-14 a}<7-a$
$\Rightarrow 58-14 a<(7-a)^2 \Rightarrow a^2-9>0$
$\Rightarrow a \in(-\infty,-3) \cup(3, \infty) \& a<\frac{29}{7}$
$\therefore a \in(-\infty,-3) \cup\left(3, \frac{29}{7}\right)$