Question

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

• A. $(-\infty ,-3)$
• B. $(-\infty ,0]$
• C. $[-3,0]$
• D. $[-3,\infty )$

Answer 1

The correct option is A $(-\infty ,-3)$

$\frac{f\left(x\right)}{x}=3\times (a+2)\times x^2-6\times a\times x+9\times a$

$(D<0)$

$D=\sqrt{36\times a^2-108\times a(a+2)}$

$72a(a+3)>0$

$a\in (-\infty ,-3)\cup (0,\infty )$

$(a+2)<0$

$a\in (-\infty ,-2)$

$ans=a\in (-\infty ,-3)$