JEE PYQ: Application Of Derivatives Question 28
Question 28 - 2020 (02 Sep Shift 2)
Let $f : (-1, \infty) \to \mathbb{R}$ be defined by $f(0) = 1$ and $f(x) = \frac{1}{x}\log_e(1 + x)$, $x \neq 0$. Then the function $f$:
(1) decreases in $(-1, 0)$ and increases in $(0, \infty)$
(2) increases in $(-1, \infty)$
(3) increases in $(-1, 0)$ and decreases in $(0, \infty)$
(4) decreases in $(-1, \infty)$
Show Answer
Answer: (4)
Solution
$f’(x) = \frac{x/(1+x) - \ln(1+x)}{x^2} = \frac{x - (1+x)\ln(1+x)}{x^2(1+x)} < 0$ for all $x \in (-1,\infty) \setminus {0}$. So $f$ is decreasing on $(-1, \infty)$.
BETA
