In the image, you're asked to find the absolute maximum and absolute minimum of a function $y = f(x)$ from its graph, along with any local maxima or minima. The graph appears to show several points of interest.

Steps for identifying extrema:

1. Identify points on the graph: The function graph shows the following key points:

   • $(2, 7)$
   • $(5, 4)$
   • $(6.5, 2)$
   • $(4, 3)$
   • $(4.3, 1)$

2. Absolute Maximum:

   • The absolute maximum is the highest point on the graph.
   • From the provided graph, the highest $y$-value occurs at $(2, 7)$. So, the absolute maximum is 7 when $x = 2$.

3. Absolute Minimum:

   • The absolute minimum is the lowest point on the graph.
   • The lowest point seems to be $(6.5, 2)$, meaning the absolute minimum is 2 when $x = 6.5$.

4. Local maxima and minima:

   • Local maximum: The function decreases after $(2, 7)$, so this point is a local maximum.
   • Local minimum: The function increases after $(6.5, 2)$, so this point is a local minimum.

Conclusion:

• Absolute Maximum is $7$ at $x = 2$.
• Absolute Minimum is $2$ at $x = 6.5$.

You should select option A and fill in the value 7 for the absolute maximum.

Would you like more details or have any questions?

Related Questions:

1. How do you distinguish between local and absolute extrema?
2. What are critical points, and how do they relate to extrema?
3. How can you confirm if a point is a local maximum or minimum?
4. What is the difference between a continuous and discontinuous function in terms of extrema?
5. Can there be more than one absolute maximum or minimum on a graph?

Tip: Always look for turning points on the graph where the slope changes direction to identify local maxima or minima.