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The area bounded by the curve y = x4 – 2x3 + x2 + 3 with x-axis and ordinates corresponding to the minima of y is
y = x4 – 2x3 + x2 + 3
⇒ y’ = 4x3 – 6x2 + 2x
At extrema of y, y’ = 0
i.e., 4x3 – 6x2 + 2x = 0
or, 2x3 – 3x2 + x = 0
⇒ x(2x – 1)(x – 1) = 0
i.e., x = 0, x = 1/2 and x = 1 correspond to extrema of y
Now, y’’ = 12x2 – 12x + 2
y’’0 = 2 > 0 ⇒ x = 0 corresponds to a minima
y’’1/2 = -1 < 0 ⇒ x = 1/2 corresponds to a maxima
y’’1 = 2 > 0 ⇒ x = 1 corresponds to a minima
So, x = 0 and x = 1 are the ordinates we need.
So, the area A is –
A =
=
(Ans)