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 `[0,1/c]` is equal to `2/3`, then the value of `(1/c+1/(c^2))` is

Options

A. 20
B. 2
C. 6
D. 12

Answer

c

Explanation

The two curves intersect at A`((1)/(c),(1)/(c^(2)))` and O (0,0) as shown in 33.

`therefore underset(0)overset(1//c)int (y_(1)-y_(2))dx=(2)/(3)`
`implies underset(0)overset(1//c)int (x^(2)-cx^(3))dx=(2)/(3)`
`implies [(x^(3))/(3)-(cx^(4))/(4)]_(0)^(1//c)=(2)/(3)`
`=(1)/(12c^(3))-(2)/(3)impliesc=(1)/(2)`
Hence, `(1)/(c)+(1)/(c^(2))=2+4=6`.