The given function is:

$f(x) = 2x(x - 1)(x - 3)^2$

Additionally, the derived function $l(x) = f(x) \cdot (x + 1)^3$ is mentioned.

The question asks:

At the point $(3,0)$, the function changes its behavior in terms of monotonicity (increasing or decreasing). You are asked to determine the correct behavior of the function around this point.

Solution approach:

1. Analyze $f(x)$:

The function $f(x) = 2x(x-1)(x-3)^2$ is a product of a cubic and quadratic polynomial. The point $(3,0)$ is a root where $x = 3$, and the term $(x - 3)^2$ ensures this is a repeated root, meaning that at $x = 3$, the derivative of $f(x)$ will vanish (no sign change occurs here for $f(x)$).

• The function doesn't cross the x-axis at $x = 3$, but rather it just touches it because $(x - 3)$ is squared.

2. Monotonicity:

At $x = 3$, since the function doesn't cross the axis but touches it and changes its concavity, it is likely that the function exhibits a local extremum (a minimum or maximum). The function is positive to the left of 3 and non-positive to the right.

Thus, the behavior near $x = 3$ would indicate that the function is decreasing before reaching 3 and increasing after passing through it, implying that the point is likely a minimum.

Conclusion:

The correct answer should be that the function undergoes a monotonic minimum at $x = 3$, indicating a change from decreasing to increasing.

Do you need further details or have any questions?

Here are 5 follow-up questions:

1. How do you find the critical points of a function?
2. What is the significance of a repeated root in a polynomial?
3. How does the derivative help in determining the monotonicity of a function?
4. What is the difference between local maxima and minima?
5. How do you determine concavity at a point?

Tip: When analyzing the behavior of a function at a given point, always check the derivative and second derivative to assess monotonicity and concavity.