Guidance for Identifying Local Extrema from a Graph

Q14. Find the location and value of each local extremum for the function shown in the graph.

Background

Topic: Local Extrema (Maximum and Minimum) from Graphs

This question tests your ability to identify local maxima and minima by analyzing the graph of a function. Local extrema are points where the function changes direction, typically at peaks (maxima) or valleys (minima).

Key Terms and Formulas:

• Local Maximum: A point where the function value is higher than at nearby points.

• Local Minimum: A point where the function value is lower than at nearby points.

• Critical Point: Where the derivative $f'(x)$ is zero or undefined.

• First Derivative Test: Used to determine if a critical point is a local max or min by checking the sign change of $f'(x)$.

Step-by-Step Guidance

1. Examine the graph and identify points where the function changes from increasing to decreasing (local maxima) or decreasing to increasing (local minima).

2. Look for peaks (highest points in a neighborhood) and valleys (lowest points in a neighborhood) along the curve. These are typically marked by vertical dashed lines at labeled points (a, b, c, d, e, f, g, h).

3. For each labeled point, check if it is a local extremum by comparing the function value at that point to the values immediately to the left and right.

4. Record the x-values where local maxima and minima occur, but do not calculate the exact function values yet.

5. Prepare to use the graph to estimate the function values at these points, but stop before writing them down.

Try solving on your own before revealing the answer!

Final Answer:

Local maxima occur at points b, d, and e. Local minima occur at points a, c, f, g, and h. The function values at these points can be read directly from the graph.

Each extremum is identified by observing where the graph changes direction. For example, at point b, the graph goes from increasing to decreasing, indicating a local maximum.