Question

If $f(x) = \log(1 + x) - \frac{2x}{2 + x}$, then $f(x)$ is increasing in ______.

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

Hint:

Always establish the natural domain of the function before analyzing intervals of increase/decrease. Even if a derivative test suggests a larger interval, the function cannot increase where it does not mathematically exist!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are asked to find the exact interval over which the given logarithmic and rational function is strictly increasing.

Step 2: Key Formula or Approach:
A function $f(x)$ is increasing on an interval if its first derivative is strictly positive ($f'(x) > 0$) for all points in that interval.
We must also ensure the interval respects the natural domain of the original function.

Step 3: Detailed Explanation:
First, identify the domain of $f(x)$. The term $\log(1+x)$ requires $1+x > 0 \implies x > -1$.
Now, find the first derivative $f'(x)$. Use the quotient rule for the second term:
$$f'(x) = \frac{1}{1+x} - \frac{(2+x)\frac{d}{dx}(2x) - 2x\frac{d}{dx}(2+x)}{(2+x)^2}$$
$$f'(x) = \frac{1}{1+x} - \frac{(2+x)(2) - (2x)(1)}{(2+x)^2}$$
$$f'(x) = \frac{1}{1+x} - \frac{4 + 2x - 2x}{(2+x)^2}$$
$$f'(x) = \frac{1}{1+x} - \frac{4}{(2+x)^2}$$
Combine the fractions by finding a common denominator:
$$f'(x) = \frac{(2+x)^2 - 4(1+x)}{(1+x)(2+x)^2}$$
Expand the numerator:
$$f'(x) = \frac{(4 + 4x + x^2) - 4 - 4x}{(1+x)(2+x)^2}$$
$$f'(x) = \frac{x^2}{(1+x)(2+x)^2}$$
For the function to be increasing, we need $f'(x) > 0$.
Observe the parts of the derivative:
- The numerator $x^2 \ge 0$ for all real $x$.
- The term $(2+x)^2 > 0$ for all $x \neq -2$.
Therefore, the sign of $f'(x)$ depends entirely on the remaining term $(1+x)$ in the denominator.
We require $1+x > 0 \implies x > -1$.
For all $x > -1$ (except $x=0$ where $f'(0)=0$, but it's an isolated inflection point), the function is strictly increasing. This also perfectly aligns with the function's natural domain constraint.

Step 4: Final Answer:
The function is increasing in $(-1, \infty)$, matching option (a).