The image contains two problems that ask you to analyze a given function based on its graph and determine local maxima and minima. Let's address each question.

Part a:

The graph of $f(x) = x^3 - 5x$ is given, and the question asks you to provide support that the function has a local maximum at approximately $x = -2$ and a local minimum at approximately $x = 2$.

To solve this:

1. Analyze the critical points: Find the first derivative of the function to identify critical points where the slope of the tangent line is zero or undefined. $f(x) = x^3 - 5x$ Taking the derivative: $f'(x) = 3x^2 - 5$ Set the derivative equal to zero to find critical points: $3x^2 - 5 = 0 \quad \Rightarrow \quad x^2 = \frac{5}{3} \quad \Rightarrow \quad x = \pm \sqrt{\frac{5}{3}}$ Approximate values for these critical points are $x \approx 1.29$ and $x \approx -1.29$.

2. Second derivative test: Now take the second derivative to determine concavity: $f''(x) = 6x$ Evaluate at the critical points:

   • For $x \approx -1.29$, $f''(x) < 0$, indicating concave down, so this point is a local maximum.
   • For $x \approx 1.29$, $f''(x) > 0$, indicating concave up, so this point is a local minimum.

Thus, this analysis supports the presence of a local maximum at approximately $x = -2$ and a local minimum at approximately $x = 2$.

Part b:

The graph of $f(x) = x^3 - 2x^2 - x + 1$ is shown, and the question asks to determine the local minima and maxima by observing the graph.

Steps to approach:

1. Local Maxima and Minima: From the graph, look for the peaks (local maxima) and troughs (local minima). Local maxima occur where the function reaches a peak and starts decreasing, while local minima occur where the function reaches a trough and starts increasing.
2. Approximate Coordinates: Based on the graph, identify the approximate $x$-coordinates where these points occur. Look for points where the derivative of the graph (i.e., the slope of the tangent line) is zero.

Since no explicit formula or numerical steps are shown in this part of the question, visually observing these critical points on the graph will help provide the necessary support for local maxima and minima.

Would you like more details on finding specific numerical values from the graph or verifying these observations? Let me know!

Here are some follow-up questions:

1. Can you explain how the first and second derivative tests help in finding local extrema?
2. What is the significance of a critical point where the derivative is zero?
3. How does concavity relate to determining whether a critical point is a local maximum or minimum?
4. Why is it useful to observe the graph when verifying local extrema?
5. How would the presence of inflection points affect the analysis of local extrema?

Tip:

Remember that local extrema occur at critical points where the first derivative is zero, but you need the second derivative to confirm whether it's a maximum or a minimum.