Determine the Interval Where f(x) = e^x / (1 + e^(4x)) is Increasing

Math Problem Statement

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

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

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:

= \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)$$, which corresponds to option **(H)**. --- Would you like more details or have any questions? Here are 5 related questions to explore further: 1. How do we determine where a function is decreasing? 2. Can you explain the steps of the quotient rule in detail? 3. What happens to $$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.

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Calculus
Derivatives
Function Analysis
Inequalities

Quotient Rule for differentiation: (f/g)' = (g f' - f g') / g^2
Natural logarithm properties
Derivative of exponential functions: (e^x)' = e^x

Quotient Rule
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