Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when

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1 < x < 2 f\left( x ight) = 2 x^3 - 9 x^2 + 12x + 29 f'\left( x ight) = 6 x^2 - 18x + 12 = 6 \left( x^2 - 3x + 2 ight) = 6\left( x - 1 ight)\left( x - 2 ight) 	ext { For f(x) to be decreasing, we must have } f'\left( x ight) < 0 \Rightarrow 6\left( x - 1 ight)\left( x - 2 ight) < 0 \Rightarrow \left( x - 1 ight)\left( x - 2 ight) < 0 \left[ 	ext { Since }6 > 0, 6\left( x - 1 ight)\left( x - 2 ight) < 0 \Rightarrow \left( x - 1 ight)\left( x - 2 ight) < 0 ight] \Rightarrow 1 < x < 2 	ext { So,f(x) is decreasing for }1 < x < 2 .

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