Question

If the function $f\left(x\right)=2x^3-\left(m-9\right)x^2+2mx$ increases in the interval $\left(0,\infty \right)$ and $m\in \left(0,9k\right),$ then the maximum value of k is $\_\_\_\_\_\_\_$

Answer 1

$f\prime \left(x\right)=6x^2-2(m-9)x+2mf\prime \left(x\right)>0....(x>0){6x}^2-2(m-9)x+2m>0D<04(m-9{)}^2-4\times (2m)\times 6<04(m^2+81-18m)-48m<04m^2-72m-48m+324<04m^2-120m+324<0m^2-30m+81<0(m-27\left)\right(m-3)<0m\in (3,27)9k=27k=3$