Question

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

• A. $(-\infty ,-3]$
• B. $\left(-\infty ,-3\right)$
• C. $\left(-\infty ,-2\right)$
• D. $(-\infty ,-3]\cup [0,\infty )$

Answer 1

The correct option is B $\left(-\infty ,-3\right)$

decreasing or strictly decreasing => $f^{\prime }\left(x\right)<0$

monotonically decreasing => $f^{\prime }\left(x\right)<=0$.

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

Given $f^{\prime }\left(x\right)<0$ => $a+2<0,(6a{)}^2-4(3(a+2)\left)\right(9a)<0$ (from quadratic equations graph concept)

after simplifying $aϵ(-\infty ,-3)$