This is an analysis of a function $f(x)$ to find its maximum and minimum values over an interval by evaluating it at certain key points. Here’s a breakdown:

Steps Explained

1. Evaluating the Function at Key Points:

   • The function $f(x)$ is evaluated at three points: $x = -4$, $x = 6$, and $x = 1$.
   • The values calculated are:
     • $f(-4) = -4e^4$
     • $f(6) = 6e^{-6}$
     • $f(1) = 1e^{-1} = e^{-1}$

2. Identifying Maximum and Minimum Values:

   • The absolute maximum and minimum values of $f(x)$ are determined from these calculations:
     • Absolute Maximum: $e^{-1}$ at $x = 1$.
     • Absolute Minimum: $-4e^4$ at $x = -4$.

   These represent the highest and lowest values of $f(x)$ among the points evaluated.

3. Sign Analysis in the Table:

   • The table below the notes provides a sign analysis, which shows the behavior of $f(x)$ around the points $x = -4$, $x = 1$, and $x = 6$.
     • For $x = -4$: $f(x)$ is increasing (indicated by a "+" symbol).
     • For $x = 1$: $f(x)$ has a zero derivative, indicating a potential maximum or minimum (a critical point).
     • For $x = 6$: $f(x)$ is decreasing (indicated by a "-" symbol).
   • The arrows indicate the direction of $f(x)$ (upward or downward) as it moves through these points.

Summary

This analysis finds the absolute maximum and minimum values of $f(x)$ at specific points by evaluating the function and analyzing its behavior around these points.

Would you like further details or have any questions?

Here are five related questions you might find interesting:

1. What is the significance of using $e$ in these calculations?
2. How would you find other critical points if they existed?
3. Why is the derivative zero at a maximum or minimum?
4. Can this method be applied to any continuous function over a closed interval?
5. What is the relationship between the signs of $f(x)$ and the shape of the function graph?

Tip: When looking for absolute extrema (maximum and minimum values) on a closed interval, always evaluate the function at critical points and the endpoints.