Question

The exhaustive set of values of m for which $f\left(x\right)=\frac{x^3}{3}+(m-1)x^2+(m+5)x+7$ is increasing function $\forall x>0$ is $[K,\infty )$ then the value of K is _________ .

Answer 1

$f\left(x\right)=\frac{x^3}{3}+(m-1)x^2+(m+5)x+7$
$f^{\prime }\left(x\right)=x^2+2(m-1)x+(m+5)$
Given $f\left(x\right)$ is increasing function for all $x>0$
$\Rightarrow m-1>0$ and $m+5>0$
$\Rightarrow m>1\therefore K=1$