Question

The interval in which the function $2x^3 + 15$ increases less rapidly than the function $9x^2 - 12x$, is

• A. $(- \infty , 1)$
• B. (1, 2)
• C. $(2, \infty )$
• D. None of these

Answer 1

Answer: (B)

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