Mathematics - Application of Derivative Question with Solution | TestHub

Let $f(x)=2x^3+15$ and $g(x)=9x^2-12x$ then ${\mathrm{f}}^{\prime }(\mathrm{x})=6{\mathrm{x}}^2\forall \mathrm{x}\in \mathrm{R}$ $\therefore \mathrm{f}(\mathrm{x})$ is increasing function $\forall \mathrm{x}\in \mathrm{R}$ $$\text{ Also, }{g}^{\prime }(x)>0\Rightarrow 18x-12>0\Rightarrow x>\frac{2}{3}$$ Thus, $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ both increases for $\mathrm{x}>\frac{2}{3}$ $$\text{ Let }F(x)=f(x)-g(x),F^{\prime }(x)<0$$ $(\because \mathrm{f}(\mathrm{x})$ increases less rapidly than the function $\mathrm{g}(\mathrm{x}))$ $$\Rightarrow 6{\mathrm{x}}^2-18\mathrm{x}+12<0\Rightarrow 1<\mathrm{x}<2$$