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To determine the intervals on which the function \( f(x) = \frac{e^x}{1 + e^{4x}} \) is increasing, we will follow these steps:

### 1. Find the derivative of \( f(x) \).

The first derivative \( f'(x) \) will help identify where the function is increasing (i.e., where \( f'(x) > 0 \)).

Given:
\[
f(x) = \frac{e^x}{1 + e^{4x}}
\]

Use the quotient rule:
\[
f'(x) = \frac{(1 + e^{4x})(e^x)' - (e^x)(1 + e^{4x})'}{(1 + e^{4x})^2}
\]

Where:
\[
(e^x)' = e^x \quad \text{and} \quad (1 + e^{4x})' = 4e^{4x}
\]

Thus:
\[
f'(x) = \frac{(1 + e^{4x})(e^x) - (e^x)(4e^{4x})}{(1 + e^{4x})^2}
\]
Simplifying:
\[
f'(x) = \frac{e^x(1 + e^{4x} - 4e^{4x})}{(1 + e^{4x})^2}
= \frac{e^x(1 - 3e^{4x})}{(1 + e^{4x})^2}
\]

### 2. Solve \( f'(x) > 0 \).

We want to find when the numerator is positive, as the denominator is always positive:
\[
e^x(1 - 3e^{4x}) > 0
\]

Since \( e^x > 0 \) for all \( x \), we focus on solving:
\[
1 - 3e^{4x} > 0
\]

\[
1 > 3e^{4x} \quad \Rightarrow \quad \frac{1}{3} > e^{4x}
\]

Taking the natural logarithm of both sides:
\[
\ln\left(\frac{1}{3}\right) > 4x
\]

\[
x < \frac{1}{4} \ln\left(\frac{1}{3}\right) = -\frac{1}{4} \ln 3
\]

Thus, \( f(x) \) is increasing when \( x < -\frac{1}{4} \ln 3 \).

### 3. Conclusion:

The function \( f(x) \) is increasing on the interval \( (-\infty, -\frac{1}{4} \ln 3) \), which corresponds to option **(H)**.

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Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How do we determine where a function is decreasing?
2. Can you explain the steps of the quotient rule in detail?
3. What happens to \( f(x) \) at \( x = -\frac{1}{4} \ln 3 \)?
4. Why is the denominator of \( f'(x) \) always positive?
5. How do logarithmic properties help solve inequalities like this one?

**Tip:** When using the quotient rule, always simplify the numerator before dividing by the denominator to avoid errors.

Let f(x) = e^x / (1 + e^(4x)). On what interval is f increasing?

Find out how to solve the problem of determining the intervals on which f(x) = e^x / (1 + e^(4x)) is increasing using the quotient rule and derivative tests. This calculus problem covers key concepts like differentiating exponential functions and analyzing function behavior through inequalities.

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