The graph in the image shows three key points:

1. $(2, 7)$
2. $(4, 3)$
3. $(5, 4)$
4. $(6, 2)$

We need to find the absolute maximum and minimum values for the function $y = f(x)$, as well as any local extrema if they exist.

• Absolute Maximum: The highest point on the graph is at $(2, 7)$, so the absolute maximum value is 7.

• Absolute Minimum: The lowest point on the graph is at $(6, 2)$, so the absolute minimum value is 2.

• Local Maxima and Minima:

  • The local maximum occurs at $(2, 7)$ and $(5, 4)$, as these points are higher than the surrounding points.
  • The local minimum occurs at $(4, 3)$, as this point is lower than its immediate surroundings.

Therefore, the correct answer is to fill in:

• Absolute maximum: $7$ (at $x = 2$).

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the definition of a local maximum in calculus?
2. How do you find absolute extrema from a graph?
3. What is the difference between absolute and local extrema?
4. Can there be more than one absolute maximum in a function?
5. How does the first derivative help in identifying extrema?

Tip: Always check both endpoints and critical points when determining absolute maxima and minima.