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The shaded portion in the graph above is represented by


Question

The shaded portion in the graph above is represented by

Options

A) y + x - x3 ≥ 0, y - x ≤ 0

B) y - x + x3 ≥ 0, y - x ≤ 0

C) y + x - x3 ≤ 0, y + x ≥ 0

D) y - x + x3 ≤ 0, y + x ≥ 0

The correct answer is A.

Explanation:

y + x - x2 ≥ 0
y = x3 - x
on x axis, y = 0
∴x2 - x = 0
x(x2 - 1) = 0
x = 0 or x2 - 1 = 0
x2 = 1
x = +/-1
∴ x = -1, 0 and 1 which are the roots of the equation y + x - x2 ≥ 0
Also y - x ≤ 0
=> y ≤ 0 + x
∴ the region where y ≤ 0

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Dicussion (1)

  • y + x - x2 ≥ 0
    y = x3 - x
    on x axis, y = 0
    ∴x2 - x = 0
    x(x2 - 1) = 0
    x = 0 or x2 - 1 = 0
    x2 = 1
    x = +/-1
    ∴ x = -1, 0 and 1 which are the roots of the equation y + x - x2 ≥ 0
    Also y - x ≤ 0
    => y ≤ 0 + x
    ∴ the region where y ≤ 0

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