Identify the stationary points of the function  y=f(x)=x^3-3x+2

To find the stationary points of the function \( y = f(x) = x^3 - 3x + 2 \), we need to:

1. Differentiate the function to find \( f'(x) \).
2. Set \( f'(x) = 0 \) to find the critical points (where the derivative equals zero).
3. Solve the equation to find the values of \( x \) for which the slope is zero (stationary points).
4. Verify the nature of these stationary points using the second derivative or by analyzing the sign changes of the first derivative.

### Step 1: Find the first derivative
The first derivative of the function \( f(x) = x^3 - 3x + 2 \) is:

\[
f'(x) = 3x^2 - 3
\]

### Step 2: Set the derivative equal to zero
To find the stationary points, we set \( f'(x) = 0 \):

\[
3x^2 - 3 = 0
\]

### Step 3: Solve for \( x \)
Solving the equation:

\[
3(x^2 - 1) = 0 \\
x^2 - 1 = 0 \\
(x - 1)(x + 1) = 0
\]

Thus, \( x = 1 \) or \( x = -1 \).

### Step 4: Verify the nature of the stationary points
To determine the nature of the stationary points, we can use the second derivative test. The second derivative of \( f(x) \) is:

\[
f''(x) = 6x
\]

- For \( x = 1 \):
  \[
  f''(1) = 6(1) = 6 > 0
  \]
  Since \( f''(1) > 0 \), the point \( x = 1 \) is a local minimum.

- For \( x = -1 \):
  \[
  f''(-1) = 6(-1) = -6 < 0
  \]
  Since \( f''(-1) < 0 \), the point \( x = -1 \) is a local maximum.

### Stationary Points:
- \( x = 1 \) is a local minimum.
- \( x = -1 \) is a local maximum.

If you substitute these values back into the original function \( f(x) \):

- At \( x = 1 \): \( f(1) = 1^3 - 3(1) + 2 = 0 \)
- At \( x = -1 \): \( f(-1) = (-1)^3 - 3(-1) + 2 = 4 \)

Thus, the stationary points are:
- \( (1, 0) \) (local minimum)
- \( (-1, 4) \) (local maximum)

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### Related Questions:
1. How do we determine if a point is an inflection point?
2. What is the significance of the second derivative in classifying stationary points?
3. Can stationary points also be points of inflection?
4. How does the behavior of a cubic function generally differ from that of a quadratic?
5. How does the third derivative of a function relate to its concavity?

### Tip:
Always check the sign of the second derivative around stationary points to confirm whether they are maxima, minima, or points of inflection.

Learn how to identify stationary points of the function y = x^3 - 3x + 2 using calculus. This problem covers differentiation, critical points, and the second derivative test to classify local maxima and minima. Suitable for advanced high school and undergraduate students.

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