To determine the absolute maximum and absolute minimum of the function $y = f(x)$ based on the graph provided, we will examine all critical points and endpoints.

Given Points:

• $(0, 3)$
• $(2, 1)$
• $(5, 6)$
• $(7, 2)$

Step 1: Identify Critical Points and Endpoints

• The endpoints are $(0, 3)$ and $(7, 2)$.
• The critical points are the local minimum at $(2, 1)$ and the local maximum at $(5, 6)$.

Step 2: Evaluate the Function Values

• $f(0) = 3$
• $f(2) = 1$
• $f(5) = 6$
• $f(7) = 2$

Step 3: Determine the Absolute Maximum and Minimum

• Absolute Maximum: The highest $y$-value is 6 at $(5, 6)$.
• Absolute Minimum: The lowest $y$-value is 1 at $(2, 1)$.

Final Answer

• Absolute Maximum: $6$ at $x = 5$
• Absolute Minimum: $1$ at $x = 2$

Would you like further details or have any questions about this solution?

Related Questions

1. How do we determine critical points analytically for functions?
2. What is the difference between a local and an absolute extremum?
3. How would the process change if the graph had more critical points?
4. Can absolute extrema occur at endpoints in all cases?
5. How can you confirm absolute maxima or minima using calculus?

Tip

Always check both critical points and endpoints when finding absolute maxima and minima in a closed interval.