First slide

Monotonicity

Question

$Let f (x) = x e^{x (1 - x)}, then f (x) is$

Moderate

A

$increasing on [- 1 / 2, 1]$
B

$decreasing on R$
C

$increasing on R$
D

$decreasing on [- 1 / 2, 1]$

Solution

$\begin{array}{l} f^{'} (x) = e^{x (1 - x)} + x (1 - 2 x) e^{x (1 - x)} \\ = (1 + x - 2 x^{2}) e^{x (1 - x)} \\ = - (x - 1) (2 x + 1) e^{x (1 - x)} \end{array}$
$\begin{array}{l} Since e^{x (1 - x)} > 0 for all x, so f^{'} (x) > 0 if and only if \\ (x - 1) (2 x + 1) < 0 i.e. - 1 / 2 = min (1, - 1 / 2) < x < max (1, - 1 / 2) = 1 \end{array}$
$Thus f increases on [- 1 / 2, 1]$

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