The function f(x) = x x is increasing in the interval - Vidyadip

MCQ

The function $f(x) = {{\log x} \over x}$ is increasing in the interval

A
$(1,\,2e)$
✓
$(0,e)$
C
$(2, 2e)$
D
$(1/e, 2e)$

✓

Answer

Correct option: B.

$(0,e)$

b
(b) $f(x) = \frac{{\log x}}{x}$

$f'(x) = \frac{1}{{{x^2}}} - \frac{{\log x}}{{{x^2}}} = \frac{{1 - \log x}}{{{x^2}}}$

For $f(x)$ to be increasing, $f'(x) > 0$

==> $1 - \log x > 0$ ==> $1 > \log x$ ==> $e > x$

$f(x)$ is increasing in the interval $(0,\,e)$.

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