To analyze the given graph and determine the local and absolute extrema, let’s examine the key points along the curve.

Observations

• The function $y = f(x)$ has four critical x-values labeled $a$, $q$, $r$, and $b$.
• The curve shows peaks and troughs, indicating points where there might be local maxima and minima.

Identifying Extrema

1. Local Maximum:

   • There is a peak between $q$ and $b$, which suggests a local maximum at $x = r$.

2. Local Minimum:

   • The curve dips between $a$ and $r$, indicating a local minimum at $x = q$.

3. Absolute Maximum:

   • The highest point on the graph is at $x = r$, so the absolute maximum also occurs here.

4. Absolute Minimum:

   • The lowest point on the curve, based on the visible region, is at $x = q$. Therefore, the absolute minimum occurs at $x = q$.

Summary of Extrema

• Local Maximum: $x = r$
• Local Minimum: $x = q$
• Absolute Maximum: $x = r$
• Absolute Minimum: $x = q$

Would you like further details on how to identify extrema or any other aspect of this graph?

Further Questions:

1. What is the general approach to finding extrema on a continuous function over a closed interval?
2. How would we confirm if a point is a local or absolute maximum using calculus?
3. Can we determine if $f(x)$ is continuous and differentiable based solely on this graph?
4. How would we find the derivative of $f(x)$ if the function's formula were known?
5. In what scenarios could a critical point be neither a maximum nor a minimum?

Tip:

When analyzing a graph for extrema, always check both endpoints if the interval is closed, as absolute extrema can occur there too.