Question

Let $f:(−1,∞)→R$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x}{\log }_e(1+x),xî€=0.$ Then the function $f$

• A. decreases in $(−1,∞)$
• B. decreases in (-1,0) and increases in $(0,∞)$
• C. increases in $(−1,∞)$
• D. increases in (-1,0) and decreases in $(0,∞)$

Answer 1

Answer: (A)

$f^{′}(x)=\frac{\frac{x}{1+x}−ℓn(1+x)}{x^2}$
$=\frac{x−(1+x)ℓn(1+x)}{x^2(1+x)}$
Suppose $h(x)=x−(1+x)ℓn(1+x)$
$⇒h^{′}(x)=1−ℓn(1+x)−1=−\ln (1+x)$
$h^{′}(x)>0,∀x∈(−1,0)$
$h^{′}(x)<0,∀x∈(0,∞)$
$h(0)=0⇒h^{′}(x)<0∀x∈(−1,∞)$
$⇒f^{′}(x)<0∀x∈(−1,∞)$
$⇒f(x)$ is a decreasing function for all $x∈(−1,∞)$