SATHEE: Application of Derivatives 2 Question 20

Application of Derivatives 2 Question 20

####20. The set of all $x$ for which $\log (1 + x) \leq x$ is equal to ….. .

(1987, 2M)

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Answer:

Correct Answer: 20. (c)

Solution:

Let $f (x) = \log (1 + x) - x$

$\Rightarrow$	$f^{'} (x) = \frac{1}{1 + x}$	$- \frac{x}{1 + x}$
$\Rightarrow$	$f^{'} (x) > 0$
when	$- 1 < x < 0$
and	$f^{'} (x) < 0$
when	$x > 0$

$∴ f (x)$ is increasing for $- 1 < x < 0$ .

$\begin{array}{lrl} \Rightarrow & f (x) & < f (0) \\ \Rightarrow & \log (1 + x) & < x \end{array}$

Again, $f (x)$ is decreasing for $x > 0$ .

$\begin{array}{ll} \Rightarrow & f (x) < f (0) \\ \Rightarrow & \log (1 + x) < x \\ ∴ & \log (1 + x) \leq x, \forall x > - 1 \end{array}$

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