Q69 / 125 · BITSAT 2014 · Mathematics
The interval in which the function $2 \mathrm{x}^{3}+15$ increases less rapidly than the function $9 x^{2}-12 x,$ is $-$
- A$(-\infty, 1)$
- Correct answerB(1,2)
- C$(2, \infty)$
- DNone of these
Solution
Let $f(x)=2 x^{3}+15$ and $g(x)=9 x^{2}-12 x$ then $\mathrm{f}^{\prime}(\mathrm{x})=6 \mathrm{x}^{2} \forall \mathrm{x} \in \mathrm{R}$
$\therefore \mathrm{f}(\mathrm{x})$ is increasing function $\forall \mathrm{x} \in \mathrm{R}$
$$
\text { Also, } g^{\prime}(x)>0 \Rightarrow 18 x-12>0 \Rightarrow x>\frac{2}{3}
$$
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 $\mathrm{g}(\mathrm{x}))$
$$
\Rightarrow 6 \mathrm{x}^{2}-18 \mathrm{x}+12<0 \Rightarrow 1<\mathrm{x}<2
$$
Hence, the correct answer is option: 'B'.