Question

The graph of $f ( x )= x ^2$ and $g ( x )= cx ^3$ intersect at two points. If the area of the region over the interval $\left[0, \frac{1}{ c }\right]$ is equal to $\frac{2}{3}$, then the value of $\left(\frac{1}{ c }+\frac{1}{ c ^2}\right)$, is

• A. 20
• B. 2
• C. 6
• D. 12

Answer 1

Answer: (C)

Obviously for $c \in(0,1), f ( x )$ lies obove the $g ( x )$
also $x ^2= cx ^3 \Rightarrow x =0$ or $x =\frac{1}{ c }$
hence $\int\limits_0^{1 / c }\left( x ^2-c x^3\right) dx =\frac{1}{12 c ^3}$

or $ \frac{1}{12 c ^3}=\frac{2}{3} \Rightarrow c ^3=\frac{1}{8} \Rightarrow c =\frac{1}{2}$
Hence, $\left(\frac{1}{ c }+\frac{1}{ c ^2}\right)=2+4=6$