Question

Find all maxima and minima of the function `y = x(x-1)^2` for `0<=x<=2` Also, determine the area bounded by the curve `y = x (x-1)^2`, the X-axis and the line x = 2 .

Answer 1

Correct Answer - `(y_(max)=(4)/(27),y_(max)=0, (10)/(3) " sq unit")`
Given, `y=x(x-1)^(2)`
`rArr (dy)/(dx) = x *2(x-1)+(x-1)^(2)`

`=(x-1)*(2x+x-1)`
`=(x-1)(3x-1)`

` therefore ` Maximum at `x = 1//3`
`y_(max) = (1)/(3) (-(2)/(3))^(2) = (4)/(27)`
Minimum at `x = 1`
`y_(max) =0`
Now, to find the area bounded by the curve `y=x(x-1)^(2),`
the Y-axis and line `x =2`.

` therefore " Required area" = "Area of square " OABC-int_(0)^(2) y dx`
`=2xx2 -int_(0)^(2) x(x-1)^(2)dx`
`=4-[[(x(x-1)^(3))/(3)]_(0)^(2)-(1)/(3) int_(0)^(2) (x-1)^(3) *1 dx]`
`=4-[(x)/(3)(x-1)^(3)-((x-1)^(4))/(12)]_(0)^(2)`
`=4-[(2)/(3)-(1)/(12)+(1)/(12)]=(10)/(3)` sq units