Question

The values ${}^{\prime }{a}^{\prime }$ for which the function $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$ decreases for all real values of $x$ is

• A. $a\le -2$
• B. $a\ge -2$
• C. $a\le -3$
• D. $-3\le a\le -2$

Answer 1

The correct option is C $a\le -3$

Given $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$ decreases for all real values of $x$

$⟹f^{\prime }\left(x\right)<0⟹3(a+2)x^2-6ax+9a<0$

$⟹discriminant\le 0⟹(-6a{)}^2-4(3\left)\right(a+2\left)\right(9\left)\right(a)\le 0$

$⟹a(a+3)\ge 0⟹a\le -3$