Question

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

• A. - 4
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A), (D)

$\Theta f(x)=x^3+3(a-7)x^2+3\left(a^2-9\right)x-1$
$\Rightarrow f^{\prime }(x)=3x^2+6(a-7)x+3\left(a^2-9\right)$
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)$D>0\Rightarrow 4(a-7{)}^2-4\left(a^2-9\right)>0$
$\Rightarrow -14a+58>0\Rightarrow a<\frac{29}{7}$
(ii) $a^2-9>0\Rightarrow a<-3$ or $a>3$
(iii) $\frac{-2(a-7)}{2\cdot 1}>0\Rightarrow a<7$
$\therefore a\in (-\infty ,-3)\cup \left(3,\frac{29}{4}\right)$