The function f(x) = x2 e−x is monotonic increasing when

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0 < x < 2 f\left( x ight) = x^2 e^{- x} f'\left( x ight) = 2x e^{- x} - x^2 e^{- x} = e^{- x} x\left( 2 - x ight) ext { For f(x) to be monotonic increasing, we must have } f'\left( x ight) > 0 \Rightarrow e^{- x} x\left( 2 - x ight) > 0 \left[ \because e^{- x} > 0 ight] \Rightarrow x\left( 2 - x ight) > 0 \Rightarrow x\left( x - 2 ight) < 0 \Rightarrow 0 < x < 2

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