Question

Find the values of $k$ for which the function $f\left(x\right)=(k+2)x^3-3k{x}^2+9kx-1$ decreases for all real values of $x$

Answer 1

$f=(K+2)x^3-3K{x}^2+9Kx-1$

$f=3(K+2)x^2-6Kx+9K\le 0$ (if $f$ is decreasing $\forall x$)

$(K+2)x^2-2Kx+3K\le 0$

$\Rightarrow K+2<0$ and $D\le 0$

$K<-2$ ....(1) $\Downarrow$

$4K^2-12K^2-24K\le 0$

$4K^2-12K^2-24K\le 0$

$-8K^2-24K\le 0$

$K^2+3K\ge 0$

$\left(K\right)(K+3)\ge 0$

$\Rightarrow K\in (-\infty ,-3]\cup [0,\infty )$ ...... (2)

$\left(1\right)\cap \left(2\right)$

$\Rightarrow K\in (-8,\infty ,-2)$