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Question
The function f(x) = x2 e−x is monotonic increasing when
Options
x ∈ R − [0, 2]
0 < x < 2
2 < x < ∞
x < 0
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Solution
0 < x < 2
\[f\left( x \right) = x^2 e^{- x} \]
\[f'\left( x \right) = 2x e^{- x} - x^2 e^{- x} \]
\[ = e^{- x} x\left( 2 - x \right)\]
\[\text { For f(x) to be monotonic increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow e^{- x} x\left( 2 - x \right) > 0 \left[ \because e^{- x} > 0 \right]\]
\[ \Rightarrow x\left( 2 - x \right) > 0\]
\[ \Rightarrow x\left( x - 2 \right) < 0\]
\[ \Rightarrow 0 < x < 2\]
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