First slide

Monotonicity

Question

If $f (x) = {xe}^{x (x - 1)}$ , then f(x) is

Easy

A

increasing on [-½,1]
B

decreasing on R
C

increasing on R
D

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

Solution

$f (x) = {xe}^{x (1 - x)}$
or $f^{'} (x) = e^{x (1 - x)} + (1 - 2 x) {xe}^{x (1 - x)} \begin{aligned} = - e^{x (1 - x)} (2 x^{2} - x - 1) \\ = - e^{x (1 - x)} (2 x + 1) (x - 1) \end{aligned}$
Sign scheme of f '(x) is as follows:
Thus, g(x) is increasing in [-½,1].

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