To find the polynomial function $f(x)$ with the given characteristics, let's break down each of the conditions.

Given Characteristics:

1. x-intercepts of -3, 1, and 5: This means that $f(x)$ has roots at these values, so the factors corresponding to these roots will be $(x + 3)$, $(x - 1)$, and $(x - 5)$.

2. Local maximum at $x = 1$: Since $x = 1$ is a local maximum, the multiplicity of the root at $x = 1$ must be even (so that the graph touches the x-axis and then turns back). This suggests the factor $(x - 1)$ must be squared: $(x - 1)^2$.

3. Local minimum at $x = -2$ and $x = 4$: Local minima typically correspond to factors with odd multiplicity. The factors corresponding to $x = -2$ and $x = 4$ should be linear (i.e., first-degree factors), so these points correspond to roots of multiplicity 1.

Step-by-step construction:

• We know that the polynomial must have the following factors:
  • $(x + 3)$ (from the root at $x = -3$),
  • $(x - 1)^2$ (from the local maximum at $x = 1$),
  • $(x + 2)$ (from the root at $x = -2$, local minimum),
  • $(x - 4)$ (from the root at $x = 4$, local minimum).

Therefore, the polynomial function will look like this:

$f(x) = k(x + 3)(x - 1)^2(x + 2)(x - 4)$

Where $k$ is a constant that determines the overall vertical stretch or shrink of the graph. This constant can be determined if additional information, like the function's value at a specific point, is provided. Since that information is not given, we will leave $k$ as an arbitrary constant for now.

Final Answer:

The polynomial function is:

$f(x) = k(x + 3)(x - 1)^2(x + 2)(x - 4)$

Do you want more details on graphing this polynomial, or how to determine the constant $k$?

Here are some related questions:

1. How would the graph of this polynomial look like?
2. What would the derivative of this polynomial look like?
3. How do we determine the value of $k$ using additional conditions?
4. What are the end behaviors of the graph based on the polynomial’s degree?
5. How does the multiplicity of the roots affect the graph near the x-intercepts?

Tip: To determine the constant $k$, use a specific point on the graph (if available), and substitute the coordinates into the equation.