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The graph shows a parabola opening downward, which means the function \( f(x) \) has a local maximum at its vertex.

To solve:

### (a) All local maximum values of \( f \):
The local maximum occurs at the highest point of the graph. By looking at the graph, the vertex is at \( (0, 4) \). Therefore, the local maximum value of \( f(x) \) is:
\[
\boxed{4}
\]

### (b) All values at which \( f \) has a local maximum:
The x-coordinate of the vertex where the local maximum occurs is \( x = 0 \). Therefore, the value at which \( f \) has a local maximum is:
\[
\boxed{0}
\]

Would you like more details on how to find local extrema from graphs, or any further questions on this?

Here are five related questions to further explore:
1. How do you find the local minima on a graph?
2. What is the difference between a global maximum and a local maximum?
3. Can a function have more than one local maximum or minimum?
4. How do you determine if a point is a local maximum using derivatives?
5. What role does the second derivative test play in finding local maxima?

**Tip**: When identifying local maxima or minima from a graph, focus on the turning points where the graph changes direction.

Finding local maxima and minima of a function given the graph

Learn how to find the local maximum and minimum of a quadratic function by analyzing its graph. This detailed example walks you through identifying the vertex and understanding the behavior of quadratic functions. Suitable for high school students.

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