Question

If \(f(x) = log(1+x)-\frac{x}{1+x}\), then the values of \(x\) for which \(f(x)\) is monotonically increasing and monotonically decreasing are respectively.....

• A. \((-\infty ,0),(0,\infty )\)
• B. \((0,\infty ),(-\infty ,0)\)
• C. \((-1,0),(0,\infty )\)
• D. \((0,\infty ),(-1,0)\)

Hint:

Differentiate and study the sign on the domain \(x>-1\).

Answer 1

The Correct Option is D

Step 1: Simplify the derivative
Combine the fractions: $\dfrac{(1+x)-1}{(1+x)^2}=\dfrac{x}{(1+x)^2}$.

Step 2: Check the sign
Positive for $x>0$ and negative for $-1<x<0$. Option (D) lists these in the asked order.

Final Answer:
Option (D) matches the intervals. \[ \boxed{(0,\infty),(-1,0)} \]