Determine whether the f(x) = x3 – x function is strictly monotonic

1.	Determine whether the f(x) = x3 – x function is strictly monotonic on the indicated interval.(-1, 0)(-1, -1/2)(-1, 1)
Answer» (x) = x³ -x ⇒ f'(x) = 3x² – 1 ⇒ f'(x) = 0 ⇒ 3x² – 1 = 0 ⇒ x = ± \(\frac{1}{\sqrt3}\) This turning point divides the domain into the intervals (-∞, \(\frac{1}{\sqrt3}\)); (-\(\frac{1}{\sqrt3}\), \(\frac{1}{\sqrt3}\)); (\(\frac{1}{\sqrt3}\), ∞). Interval (-1, 0), f'(x) changes sign. So not monotonic. Interval (-1, -1/2), f'(x) > 0 strictly monotonic. lnterval (-1, 1) not monotonic

Determine whether the f(x) = x3 – x function is strictly monotonic on the indicated interval.(-1, 0)(-1, -1/2)(-1, 1)

Answer»

(x) = x³ -x ⇒ f'(x) = 3x² – 1
⇒ f'(x) = 0 ⇒ 3x² – 1 = 0 ⇒ x = ± \(\frac{1}{\sqrt3}\)
This turning point divides the domain into the intervals (-∞, \(\frac{1}{\sqrt3}\)); (-\(\frac{1}{\sqrt3}\), \(\frac{1}{\sqrt3}\)); (\(\frac{1}{\sqrt3}\), ∞).

Interval (-1, 0), f'(x) changes sign. So not monotonic.
Interval (-1, -1/2), f'(x) > 0 strictly monotonic.
lnterval (-1, 1) not monotonic

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Determine whether the f(x) = x3 – x function is strictly monotonic on the indicated interval.(-1, 0)(-1, -1/2)(-1, 1)

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