Question

If the function $f\left(x\right)=x^3+3(a-7)x^2+3(a^2-9)x-1$ has a positive point of maximum, then

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

Answer 1

The correct option is B $a\in (-\infty ,-3)\cup (3,\frac{29}{7})$

$f\left(x\right)=x^3+3(a-7)x^2+3(a^2-9)x-1$
$f^{\prime }\left(x\right)=3x^2+6(a-7)x+3(a^2-9)$
$D>0,f^{\prime }\left(0\right)>0$
$D>0\Rightarrow 36(a-7{)}^2-36(a^2-9)>0$
$\Rightarrow a<\frac{29}{7}$

$f^{\prime }\left(0\right)=3(a^2-9)>0$
$\Rightarrow (a-3)(a+3)>0$
$\Rightarrow a<-3$ or $a>3$
Hence $a\in (-\infty ,-3)\cup (3,\frac{29}{7})$