Question

The value of $a$ for which the function
$(a+2)x^3-3a{x}^2+9ax-1$ decreases monotonically for all real $x$, is

• A. $a<-2$
• B. $a>-2$
• C. $-3<a<0$
• D. $-\infty <a\le -3$

Answer 1

The correct option is D $-\infty <a\le -3$

$f^{\prime }\left(x\right)\ge 0$ for it to be a decreasing function.
Hence
$3(a+2)x^2-6ax+9a\ge 0$
$3\left(\right(a+2)x^2-2ax+3a)\ge 0$
Or
$D\le 0$
$4a^2-12a(a+2)\ge 0$
$4a(a-3(a+2\left)\right)\ge 0$
$4a(-6-2a)\ge 0$
$8a(-3-a)\ge 0$ Or
$8a(a+3)\le 0$
Hence
$a>0$ and $a+3<0$
OR $a<0$ and $a+3>0$
Hence
$aϵ(-\infty ,-3]$.