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The function f: R R is defined by f(x) = x − (x3)/(3). Find the monotony of f, showing clearly its variation table.

Mathematics

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Graph of f(x) = x − x³/3

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Key points of f(x) = x − x³/3
Pointxy
x-intercept−1.730
Minimum−1−0.67
x-intercept00
Maximum10.67
x-intercept1.730

Step-by-step solution

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Part 3

Part i· 2 steps

The function f:R→Rf: \mathbb{R} \to \mathbb{R} is defined by f(x)=x−x33f(x) = x - \frac{x^3}{3}. Find the monotony of ff, showing clearly its variation table.

  1. Find the first derivative of f(x)f(x).

    f(x)=x−x33f(x) = x - \frac{x^3}{3} f′(x)=ddx(x−x33)=1−3x23=1−x2f'(x) = \frac{d}{dx}\left(x - \frac{x^3}{3}\right) = 1 - \frac{3x^2}{3} = 1 - x^2

  2. Find the critical

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