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