We are given four polynomial functions, and the task is to identify which of them first increases and then decreases in the intervals $(-4, -3)$, $(-1, 2)$, and $(5, 6)$.

Step-by-step approach:

1. Function Behavior: To determine where a function increases or decreases, we need to analyze the derivative. However, a quicker way is to observe the roots and the powers of the factors of each function, as these determine the sign changes of the function.
2. Critical Points: Focus on the factors of the form $(x - a)$, as the roots and their multiplicities will tell us if the function crosses the x-axis (indicating a sign change, i.e., increasing to decreasing or vice versa).
   • If the multiplicity is odd, the function crosses the x-axis, changing its sign.
   • If the multiplicity is even, the function touches the x-axis but does not cross it, maintaining the same sign.

Interval Analysis:

We are interested in three intervals: $(-4, -3)$, $(-1, 2)$, and $(5, 6)$, which means the function should increase first and then decrease within these intervals.

Let’s examine the provided options based on their roots and multiplicities.

Option 1:

$f(x) = -\frac{1}{10000} (x + 1)^2 (x - 2) (x + 3)^2 (x + 4)^2 (5 - x) (x - 6)^2$

• Roots: $x = -1$, $x = 2$, $x = -3$, $x = -4$, $x = 5$, $x = 6$.
• Multiplicities:
  • $x = -1$: Even (so no sign change).
  • $x = 2$: Odd (sign changes here).
  • $x = -3$: Even.
  • $x = -4$: Even.
  • $x = 5$: Odd.
  • $x = 6$: Even.

This doesn't match the required behavior in the specified intervals.

Option 2:

$f(x) = \frac{1}{10000} (x + 1)^2 (x - 2) (x + 3) (x + 4) (x - 5)^2 (x - 6)^2 (x + 7)$

• Roots: $x = -1$, $x = 2$, $x = -3$, $x = -4$, $x = 5$, $x = 6$, $x = -7$.
• Multiplicities:
  • $x = -1$: Even.
  • $x = 2$: Odd (sign change).
  • $x = -3$: Odd (sign change).
  • $x = -4$: Odd (sign change).
  • $x = 5$: Even.
  • $x = 6$: Even.
  • $x = -7$: Odd.

This option seems complex but could be explored further.

Option 3:

$f(x) = -\frac{1}{10000} (x + 1)^2 (x - 2) (x + 3)^2 (x + 4)^2 (5 - x)^2 (x - 6)^2 (3 - x)$

• Roots: Similar to Option 1 but with some differences in terms and signs. Multiplicities also differ, but overall it does not seem to satisfy the given condition.

Option 4:

$f(x) = \frac{1}{10000} (x + 1)^2 (x - 2) (x + 3)^2 (x + 4) (x - 5)^2 (x - 6)^2$

• Roots: $x = -1$, $x = 2$, $x = -3$, $x = -4$, $x = 5$, $x = 6$.
• Multiplicities:
  • $x = -1$: Even.
  • $x = 2$: Odd (sign change).
  • $x = -3$: Even.
  • $x = -4$: Odd (sign change).
  • $x = 5$: Even.
  • $x = 6$: Even.

This function seems to match the behavior expected for the intervals. The sign changes align with increasing and decreasing behavior in the specified intervals.

Conclusion:

Option 4 is the correct answer because the sign changes and multiplicities match the required behavior in the intervals $(-4, -3)$, $(-1, 2)$, and $(5, 6)$.

Would you like more details on any specific part of the solution? Here are 5 questions to explore further:

1. How do multiplicities affect the sign of a polynomial function?
2. What is the role of critical points in determining increasing or decreasing behavior?
3. How can the derivative help in finding where a function increases or decreases?
4. What is the significance of even and odd multiplicities in the behavior of functions?
5. Can we graph these functions to visually confirm their behavior?

Tip: Always check the multiplicity of each root when determining the behavior of polynomial functions—this reveals whether the function crosses the axis or just touches it.