To determine the local maximum of the function $f(x) = x^4 + 4x^3 - x$ on the interval $-5 \leq x \leq 0$, we need to find the critical points by setting the derivative of $f(x)$ to zero. Let’s work through the process.

1. Find the derivative $f'(x)$: $f'(x) = \frac{d}{dx} (x^4 + 4x^3 - x) = 4x^3 + 12x^2 - 1$

2. Set the derivative equal to zero to find critical points: $4x^3 + 12x^2 - 1 = 0$

This equation matches option 1 in the provided answers, which is:

• $4x^3 + 12x^2 - 1 = 0$

To determine the local maximum within the specified domain, solving this equation will give us the critical points, which can then be analyzed to identify if they are maximum points.

Would you like more details on how to solve this equation or proceed with further analysis of the critical points?

Here are some additional questions related to this topic:

1. How do we determine whether a critical point is a maximum or minimum?
2. What role does the second derivative play in identifying local maxima and minima?
3. How can we verify if a function has a maximum or minimum on a closed interval?
4. Why is it necessary to check the endpoints in addition to the critical points in closed intervals?
5. What are some techniques for solving polynomial equations like $4x^3 + 12x^2 - 1 = 0$?

Tip: When solving for local maxima or minima, always check both critical points and endpoints within the domain for a complete analysis.