Let's begin by analyzing the two images you've provided, starting with identifying the local minima based on the first graph.

In the first graph:

• The function $f(x)$ has local minima where the curve dips and then rises again.
• Looking at the graph, local minima occur at the points where the slope changes from negative to positive, i.e., the points labeled $v$ and $u$.

Thus, the local minima occur at $x = v$ and $x = u$.

Detailed Explanation:

• A local minimum occurs at a point if the function has a smaller value than at nearby points on either side.
• In the graph, at point $v$, the curve dips down and starts increasing afterward, indicating a local minimum.
• Similarly, at point $u$, the function dips before rising again, marking another local minimum.

So, the answer for the local minima from the first graph is:

• Option A: The local minimum values occur at $x = v, u$.

Now, analyzing the second image (example provided):

• The graph describes a function $f(x)$ over the interval $[a, b]$.
• As stated in the explanation:
  • Local minima occur at $q$ and $s$.
  • Local maxima occur at $p$, $r$, and $t$.
  • The function has an absolute minimum at $b$ and an absolute maximum at $p$.

This second image illustrates the behavior of the function, with points of local extrema occurring at specific interior points of the interval.

Would you like more details on the reasoning or have any questions?

Here are 5 relative questions to further explore the topic:

1. What are the conditions for a point to be considered a local minimum or maximum?
2. How do you differentiate between local and absolute extrema?
3. Why are endpoints considered for absolute extrema in closed intervals?
4. What role do critical points play in finding extrema?
5. How would the graph change if the interval were open instead of closed?

Tip: Always check both the interior points and endpoints of an interval for absolute extrema when dealing with continuous functions on a closed interval.