To analyze the function $f(x) = x^3 - 3x + 2$, we can break down the tasks into the following steps:

1. Graph the Function: We'll use a graphing tool to plot $f(x) = x^3 - 3x + 2$ over the interval $[-2, 2]$.
2. Find Local Maximum and Minimum Values: We can approximate the local extrema by finding the derivative $f'(x)$, setting it to zero to locate critical points, and then analyzing the function's behavior around those points.
3. Determine Intervals of Increase and Decrease: By examining the sign of $f'(x)$, we can determine where $f(x)$ is increasing or decreasing.

Step 1: Compute the Derivative

The derivative of $f(x) = x^3 - 3x + 2$ is: $f'(x) = 3x^2 - 3$

Setting $f'(x) = 0$ gives: $3x^2 - 3 = 0 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1$

So, the critical points are at $x = 1$ and $x = -1$.

Step 2: Determine Local Extrema

To confirm whether these critical points are local maxima or minima, we can evaluate $f(x)$ at $x = -1$ and $x = 1$ and consider the sign of $f'(x)$ around these points.

• At $x = -1$: $f(-1) = (-1)^3 - 3(-1) + 2 = -1 + 3 + 2 = 4$

• At $x = 1$: $f(1) = (1)^3 - 3(1) + 2 = 1 - 3 + 2 = 0$

Using these results, we approximate that there is a local maximum at $(-1, 4)$ and a local minimum at $(1, 0)$.

Step 3: Intervals of Increase and Decrease

To determine the intervals where $f(x)$ is increasing or decreasing, we analyze $f'(x) = 3x^2 - 3$:

• When $x < -1$, $f'(x) > 0$ (increasing).
• When $-1 < x < 1$, $f'(x) < 0$ (decreasing).
• When $x > 1$, $f'(x) > 0$ (increasing).

Thus, $f(x)$ is:

• Increasing on $(-\infty, -1)$ and $(1, \infty)$.
• Decreasing on $(-1, 1)$.

Summary

• Local Maximum: At $(-1, 4)$
• Local Minimum: At $(1, 0)$
• Intervals of Increase: $(-\infty, -1) \cup (1, \infty)$
• Intervals of Decrease: $(-1, 1)$

Would you like a detailed graph or further explanation?

---

Follow-Up Questions

1. What is the second derivative of $f(x) = x^3 - 3x + 2$, and what does it tell us about concavity?
2. How would the behavior of $f(x)$ change if we modified the constant term?
3. What are the points of inflection for this function, if any?
4. How does the behavior of $f(x)$ compare to that of $g(x) = x^3$?
5. What effect does the coefficient of $x$ have on the local extrema of similar cubic functions?

Tip: When analyzing function behavior, check critical points using both first and second derivatives to understand the shape of the curve.