For all x (0, ,1) - Vidyadip

MCQ

For all $x \in (0,\,1)$

A
${e^x} < 1 + x$
✓
${\log _e}(1 + x) < x$
C
$\sin x > x$
D
${\log _e}x > x$

✓

Answer

Correct option: B.

${\log _e}(1 + x) < x$

b
(b) Both ${e^x}$ and $1 + x$ are increasing and $\sqrt e \ge 1 + \frac{1}{2},$ because $\sqrt e = 1.65$ nearly.

so the answer $(a)$ is not correct.

Since $\sin \frac{\pi }{6} < \frac{\pi }{6}$ because $\frac{1}{2} < \frac{{22}}{{42}}$.

So,$ (c) $ is not correct.

$\log \frac{1}{2} < \frac{1}{2}$ because $\log \frac{1}{2}$ is negative.

$\therefore $ Option $(d)$ is not correct.

Thus, by elimination $ (b)$ is correct.

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