Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.

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Function f(x) = x3 − 27x + 5 is monotonically increasing when | x | > 3. Explanation: f\left( x ight) = x^3 - 27x + 5 f'\left( x ight) = 3 x^2 - 27 = 3 \left( x^2 - 9 ight) 	ext { For f(x) to be increasing, we must have } f'\left( x ight) > 0 \Rightarrow 3 \left( x^2 - 9 ight) > 0 \Rightarrow \left( x^2 - 9 ight) > 0 \left[ 	ext { Since } 3 > 0, 3 \left( x^2 - 9 ight) > 0 \Rightarrow \left( x^2 - 9 ight) > 0| ight] \Rightarrow \left( x + 3 ight)\left( x - 3 ight) > 0 \Rightarrow x < - 3 \ or \ x > 3 \Rightarrow \left| x ight| > 3

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