1. **Problem Statement:** Analyze the polynomial function's graph and verify which statements about its characteristics are true. 2. **Key Observations from the Graph:** - The graph crosses the x-axis near $x=-2$ and $x=2$. - There are exactly two local extrema: a local maximum near $x=-1$ and a local minimum near $x=1$. - As $x \to -\infty$, $f(x) \to \infty$. - As $x \to \infty$, $f(x) \to -\infty$. - The function crosses the x-axis at $x=2$ with an odd multiplicity (since it crosses rather than just touches). - The leading coefficient is negative (due to end behavior: left end up, right end down). - The least possible degree is 5, consistent with the number of extrema and end behavior. - The range is all real numbers $(-\infty, \infty)$ because the graph extends infinitely in both vertical directions. 3. **Explanation of Statements:** - The 5th finite differences being constant negative corresponds to a degree 5 polynomial with negative leading coefficient. - The function is not even (no line symmetry about the y-axis). - The y-intercept is not negative (graph does not show negative y-intercept). - The graph does not have an absolute maximum since it goes to infinity as $x \to -\infty$. 4. **Summary of True Statements:** - As $x \to -\infty$, $f(x) \to \infty$. - The function has exactly two local extrema. - The range is $(-\infty, \infty)$. - The multiplicity of the x-intercept at $x=2$ is odd. - The least possible degree is 5. - The leading coefficient is negative. 5. **Final Answer:** The true statements are those marked with [x] in the problem.