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in Limit, continuity and differentiability by (50.2k points)

Let f(x) = 1 – x – x3. Find all real values of x satisfying the inequality 1 – f(x) – f3(x) > f(1 – 5x).

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We have f(x) = 1 – x – x3

 f'(x) = –1 – 3x2 < 0, " x ∈ R

f(x) is strictly decreasing

Now, f( f(x)) = 1 – f(x) – ( f(x))3

 f(f(x)) > f(1 – 5x)

 f(x) < (1 – 5x)

 1 – x – x3 < (1 – 5x)

 x3 – 4x > 0

 x(x2 – 4) > 0

 x(x + 2)(x – 2) > 0

 x (–2, 0) ∪ (2, )

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