To solve for the positive intervals of the function $f(x) = x^3 - 4x^2 + x + 6$, we need to follow these steps:

Step 1: Find the roots of the function

We start by solving the equation $f(x) = 0$, which means solving: $x^3 - 4x^2 + x + 6 = 0$ We can try factoring or using numerical methods (like synthetic division or a calculator) to approximate the roots.

Step 2: Analyze the sign of $f(x)$

Once we find the roots, we can determine the sign of $f(x)$ in the intervals defined by these roots. Since we are looking for the positive intervals, we want to identify where $f(x) > 0$.

Step 3: Combine intervals

From the results of Step 2, we can construct the union of intervals where the function is positive.

Let me solve the cubic equation to find the roots and determine the positive intervals.The real roots of the function $f(x) = x^3 - 4x^2 + x + 6$ are $x = -1$, $x = 2$, and $x = 3$.

Step 4: Analyze the sign of the function

The roots divide the real line into the following intervals:

• $(-\infty, -1)$
• $(-1, 2)$
• $(2, 3)$
• $(3, \infty)$

We will now check the sign of the function in each interval by selecting a test point from each.

• For $(-\infty, -1)$, choose $x = -2$.
• For $(-1, 2)$, choose $x = 0$.
• For $(2, 3)$, choose $x = 2.5$.
• For $(3, \infty)$, choose $x = 4$.

Let me calculate $f(x)$ at these test points to determine where the function is positive.The sign of the function $f(x)$ at the selected test points is:

• For $(-\infty, -1)$, at $x = -2$, $f(x) = -20$ (negative).
• For $(-1, 2)$, at $x = 0$, $f(x) = 6$ (positive).
• For $(2, 3)$, at $x = 2.5$, $f(x) = -0.875$ (negative).
• For $(3, \infty)$, at $x = 4$, $f(x) = 10$ (positive).

Step 5: Identify positive intervals

The function is positive in the following intervals:

• $(-1, 2)$
• $(3, \infty)$

Thus, the positive intervals of the function are $(-1, 2) \cup (3, \infty)$.

Final Answer:

The correct choice is $(-1, 2) \cup (3, \infty)$.

Would you like further details or have any questions?

Here are five related questions you might want to explore:

1. How do you factorize a cubic equation?
2. How do you determine the sign of a polynomial in an interval?
3. What methods can be used to solve cubic equations?
4. What is the significance of real versus complex roots?
5. How does the graph of a cubic function behave around its roots?

Tip: When analyzing polynomial functions, always remember that the sign of the function changes at its roots. This helps you quickly determine intervals of positivity or negativity.