JEE Main & Advanced Sample Paper JEE Main - Mock Test - 12

The interval in which the function \[2{{x}^{3}}+15\] increases less rapidly than the function \[9{{x}^{2}}-12x,\] is

A) \[(-\infty ,1)\]

B) \[(1,2)\]

C) \[(2,\infty )\]

D) None of these

Correct Answer: B

Solution :

Let \[f(x)=2{{x}^{3}}+15\] and \[g(x)=9{{x}^{2}}-12x\]then

\[f'(x)=6{{x}^{2}}\,\forall \,x\in R\]

\[\therefore \,\,f(x)\] is increasing function \[\forall \,x\in R\]

Also, \[g'(x)>0\Rightarrow 18x-12>0\Rightarrow x>\frac{2}{3}\]

Thus, \[f(x)\] and \[g(x)\] both increases for \[x>\frac{2}{3}\]

Let \[F(x)=f(x)-g(x),\] \[F'(x)<0\]

(\[\because \] \[f(x)\] increases less rapidly than the function g \[(x)\])

\[\Rightarrow \,\,6{{x}^{2}}-18x+12<0\]

\[\Rightarrow \,\,\,1<x<2\]