Home
Class 12
MATHS
Find all maxima and minima of the functi...

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 .

Text Solution

Verified by Experts

The correct Answer is:
`(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
Promotional Banner

Similar Questions

Explore conceptually related problems

Find the maxima and minima of the function y=x(x-1)^2,0lt=xlt=2.

Find all points of local maxima and local minima of the function f given by f(x) = x^(3) – 3x + 3.

Find all the points of local maxima and local minima of the function f given by f(x) = 2x^( 3) – 6x^(2) + 6x +5.

Find all the points of local maxima and local minima of the function f given by f(x) = 2x^(3) – 6x^(2) + 6x +5.

Determine the point of maxima and minima of the function f(x)=1/8(log)_e x-b x+x^2,x >0, where bgeq0 is constant.

Discuss the maxima and minima of the function f(x)=x^(2/3)-x^(4/3)dot Draw the graph of y=f(x) and find the range of f(x)dot

Discuss the maxima and minima of the function f(x)=x^(2/3)-x^(4/3)dot Draw the graph of y=f(x) and find the range of f(x)dot

Investigate for the maxima and minima of the function f(x)=int_1^x[2(t-1)(t-2)^3+3(t-1)^2(t-2)^2]dt

Find the local maxima and local minima, if any, of the functions. Find also the local maximum and the local minimum values, as the case may be: g(x ) =(x)/(2) +(2)/(x ) x gt 0

Find the local maxima and local minima, if any, of the functions. Find also the local maximum and the local minimum values, as the case may be: h(x) = sin x+ cos x , 0 lt x lt (pi)/(2)