Let's analyze the graph step by step to determine the following:

1. Degree of $f(x)$: The degree of the polynomial corresponds to the number of turning points plus one. Observing the graph:

   • There are 3 turning points (two peaks and one trough).
   • Therefore, the degree of the polynomial is 4 (one more than the number of turning points).

2. Leading Coefficient: The behavior of the graph at the ends (as $x \to \pm \infty$) indicates the sign of the leading coefficient:

   • The graph starts high on the left ($y \to +\infty$ as $x \to -\infty$) and ends low on the right ($y \to -\infty$ as $x \to +\infty$).
   • This implies the leading coefficient is negative.

3. Distinct Real Zeros: Count the number of x-intercepts (where the graph crosses the x-axis):

   • The graph intersects the x-axis 3 times.
   • Hence, there are 3 distinct real zeros.

4. Relative Minimums: Relative minimums occur at troughs (the lowest points in their neighborhood):

   • There is 1 relative minimum (the single trough in the graph).

Final Answer:

• Degree of $f(x)$: 4
• Leading Coefficient: Negative
• Distinct Real Zeros: 3
• Relative Minimums: 1

Would you like further explanation on any of these points or a related concept? Here are some additional questions to deepen understanding:

1. How can you identify the degree of a polynomial from its graph without counting turning points?
2. What determines the sign of the leading coefficient based on the graph's end behavior?
3. How are multiplicities of roots reflected in a polynomial graph?
4. Can there be more relative extrema than the degree minus one?
5. How would the graph change if the leading coefficient were positive?

Tip: Always observe the end behavior of the graph to deduce the sign of the leading coefficient, especially for high-degree polynomials.