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