Mathematics - Application of Derivative Question with Solution | TestHub

$x^3-9x^2+24x=-k$ graph of LHS $\mathrm{f}(\mathrm{x})={\mathrm{x}}^3-9{\mathrm{x}}^2+24\mathrm{x}$ ${\mathrm{f}}^{\prime }(\mathrm{x})=3{\mathrm{x}}^2-18\mathrm{x}+24$ ${\mathrm{f}}^{\prime }(\mathrm{x})=3\left({\mathrm{x}}^2-6\mathrm{x}+8\right)$ $=3(x-2)(x-4)$

$\mathrm{x}=2$ maxima $\mathrm{x}=4$ minima at $x=2f(x)=20$ at $x=4f(x)=16$

for three solutions $16<-\mathrm{k}<20$ $\mathrm{k}\in (-20,-16)$