Use a graphing utility to graph f left parenthesis x right parenthesis equals x cubed minus 3 x plus 2f(x)=x3−3x+2 on the interval left bracket negative 2 comma 2 right bracket[−2,2] and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing.

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?

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### 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.

Use a graphing utility to graph f(x) = x^3 - 3x + 2 on the interval [-2,2] and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing.

Explore how to graph the cubic function f(x) = x^3 - 3x + 2 over the interval [-2,2]. Learn to approximate local maxima and minima, and determine intervals of increase and decrease using derivative analysis. Ideal for students in Grades 11-12 studying calculus concepts.

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