First slide

Monotonicity

Question

$f (x) = |{xlog}_{e} x|$ monotonically decreases in

Moderate

A

$(0, 1 / e)$
B

$(1 / e, 1)$
C

$(1, \infty)$
D

$(1 / e, \infty)$

Solution

$f (x) = |{xlog}_{e} x| For g (x) = x \log_{e} x,$
$g^{'} (x) = x \frac{1}{x} + \log_{e} x = 1 + \log_{e} x$
Thus, g (x) increases for $(\frac{1}{e}, \infty)$ and decreases for $(0, \frac{1}{e})$ .

From the graph, $f (x) = |{xlog}_{e} x|$ decreases in $(0, \frac{1}{e})$ .

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