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The image you provided shows a mathematical problem asking to "Determine the intervals on which the function \( f(x) = \frac{5}{x^2} + \frac{4}{x^3} - \frac{1}{x^4} \) is increasing and decreasing."

Let's break this down step by step:

### Steps to Solve:
1. **Find the first derivative** of the function \( f(x) \). The critical points (where \( f'(x) = 0 \) or is undefined) will help determine where the function changes from increasing to decreasing (or vice versa).

2. **Determine the sign of the first derivative** \( f'(x) \) on different intervals. This will tell us if the function is increasing or decreasing in those intervals.

3. **Analyze critical points** to figure out where the function changes its behavior.

Let's begin solving it.

Given:
\[
f(x) = \frac{5}{x^2} + \frac{4}{x^3} - \frac{1}{x^4}
\]
We will differentiate this function with respect to \( x \).The first derivative of the function \( f(x) = \frac{5}{x^2} + \frac{4}{x^3} - \frac{1}{x^4} \) is:

\[
f'(x) = -\frac{10}{x^3} - \frac{12}{x^4} + \frac{4}{x^5}
\]

Now, we need to analyze this derivative to determine the intervals where the function is increasing and decreasing. This involves finding the critical points and checking the sign of the derivative in different intervals.

Next, let's solve \( f'(x) = 0 \) to find the critical points.The critical points are:

\[
x_1 = \frac{-3 + \sqrt{19}}{5}
\quad \text{and} \quad
x_2 = \frac{-3 - \sqrt{19}}{5}
\]

These critical points will help us divide the number line into intervals. Now, we will check the sign of the derivative \( f'(x) \) in each of these intervals to determine whether the function is increasing or decreasing.

Let's analyze the sign of \( f'(x) \) on intervals defined by these critical points.The sign analysis of \( f'(x) \) at the test points reveals the following:

- **Left of \( x_2 \)**: \( f'(x) > 0 \), so the function is increasing in this interval.
- **Between \( x_2 \) and \( x_1 \)**: \( f'(x) < 0 \), so the function is decreasing in this interval.
- **Right of \( x_1 \)**: \( f'(x) < 0 \), so the function is still decreasing in this interval.

### Conclusion:
- The function \( f(x) \) is **increasing** on the interval \( (-\infty, x_2) \).
- The function is **decreasing** on the intervals \( (x_2, x_1) \) and \( (x_1, \infty) \).

Would you like further details on these calculations, or any clarifications? 

Here are 5 related questions to explore:
1. How does the second derivative help in identifying concavity?
2. What do critical points tell us about local extrema?
3. How do we classify points where \( f'(x) = 0 \)?
4. What happens to the behavior of the function as \( x \) approaches infinity or negative infinity?
5. How can you verify these results graphically?

**Tip**: Always plot the function and its derivative for visual confirmation of increasing and decreasing intervals.

Determine the intervals on which the function f(x) = (5/x^2) + (4/x^3) - (1/x^4) is increasing and decreasing.

Learn how to determine the intervals on which the function f(x) = (5/x^2) + (4/x^3) - (1/x^4) is increasing and decreasing. This solution uses derivatives and critical points, applying the first derivative test.

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