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Graph of Quadratic Expression

The figure sh...

Question

The figure shows a portion of the graph $y = 2 x - 4 x^{3}$ . The line y = c is such that the areas of the regions marked I and II are equal. If a,b are the x-coordinates of A,B respectively, then a + b equals

A

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B

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C

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D

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Solution

The correct option is A

$\int_{a}^{b} (2 x - 4 x^{3}) d x = 2 (b - a) C (x^{2} - x^{4})_{a}^{b} = 2 (b - a) C (b + a) [1 - (a^{2} + b^{2})] = 2 C (a + b) ⌊ 1 - (a + b)^{2} + 2 a b ⌋ = 2 C . . . . (i) Again 2 x - 4 x^{2} = C 4 x^{3} - 2 x + C = 0 Roots, a, b, α$
$4 x^{3} + 0. x^{2} - 2 x + C = 0$

$a + b + α = 0 . . . (i i) a b + (a + b) α = - \frac{1}{2} . . . . . (i i i) 7 α^{3} = 4 α \Rightarrow α = \frac{2}{\sqrt{7}}$

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y = 2 x - 4 x^{3}

. The line y = c is such that the areas of the regions marked I and II are equal. If a,b are the x-coordinates of A,B respectively, then a + b equals

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