The area bounded by the curve y=x^4-2 x^3+x^2+3 with x -axis and ordinates corresponding to the minima of y is

Step-by-step solution

The equation of given curve is y=x^4-2 x^3+x^2+3 On differentiating w.r.t. x , we get d y d x =4 x^3-6 x^2+2 x Again, differentiating, we get d^2 y d x^2 =12 x^2-12 x+2 Put, d y d x =0 for maxima or minima. 2 x (2 x^2-3 x+1 )=0 x=0,1, 1 2 ( d^2 y d x^2 )_ (x=0) =2 Function is minimum at x=0 and ( d^2 y d x^2 )_ x=1 =12-12+2=2 Function is minimum at x=1 Also, ( d^2 y d x^2 )_ (x= 1 2 ) =-1 0 Function is maximum at x= 1 2 Required area = ₀^1 (x^4-2 x^3+x^2+3 ) d x= [ x^5 5 - 2 x^4 4 + x^3 3 +3 x ]₀^1 aligned & = 1 5