To analyze the graph provided, I'll answer each of your questions based on the visual properties of the function $f(x)$:

1. Intervals on which the function is increasing:

• A function is increasing where the slope of the curve is positive.
• From the graph, we see that the function is increasing on two intervals:
  • From $x \approx -5.5$ to $x \approx -3$ (before it reaches the local maximum).
  • From $x \approx 1$ and onwards (as the curve continues rising).

Intervals of increase: $(-5.5, -3) \cup (1, \infty)$.

2. Values of $x$ at which the local minima occur:

• A local minimum occurs where the function changes from decreasing to increasing (i.e., a "valley").
• The graph shows local minima at approximately:
  • $x \approx -5.5$ (the function has a clear valley before it starts rising again).
  • $x = 0$ (the function dips down and then starts increasing at this point).

Local minima: At $x \approx -5.5$ and $x = 0$.

3. Values of the local maxima:

• A local maximum occurs where the function changes from increasing to decreasing (i.e., a "peak").
• The graph shows a local maximum at approximately:
  • $x \approx -3$, where the curve peaks before descending again.

Local maximum value: The value of the function at this peak appears to be approximately $f(x) = 2$.

Would you like more details on any part of this explanation or a breakdown of the specific intervals?

Follow-up questions:

1. How can you verify increasing and decreasing intervals using derivatives?
2. How can the first derivative test help confirm the nature of extrema?
3. What is the significance of concavity in determining local minima and maxima?
4. How would you estimate the exact values of the local minima and maxima?
5. How could you approximate the x-intercepts or zeros of this function?

Tip:

When analyzing graphs of functions, always check for changes in slope (positive/negative) to determine intervals of increase or decrease.