QJEEM22Mae4fb: JEE Main 2022 Mathematics - Class 12 JEEM22Mae4fb Solution: The function f ( x ) = x e x ( 1 − x ) , x ∈ R , is :

Question

Verified Solution • 2025–26 Session moderate

The function $f (x) = x e^{x (1 - x)}, x \in R$ , is :

Accepted Answer

$f (x) = x e^{x (1 - x)}, x \in R$

$f^{'} (x) = x e^{x (1 - x)} . (1 - 2 x) + e^{x (1 - x)}$

$= e^{x (1 - x)} [x - 2 x^{2} + 1]$

$= - e^{x (1 - x)} [2 x^{2} - x - 1]$

$= - e^{x (1 - x)} (2 x + 1) (x - 1)$

$∴$ $f (x)$ is increasing in $(- \frac{1}{2}, 1)$ and decreasing in $(- \infty, - \frac{1}{2}) \cup (1, \infty)$

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