To find the maxima and minima of the function \( f(x) = x^3 - 3x + 11 \), we first need to follow these steps:

1. **Find the first derivative:** This will help us find the critical points by setting the derivative equal to zero.

\[
f'(x) = \frac{d}{dx}(x^3 - 3x + 11) = 3x^2 - 3
\]

2. **Set the derivative equal to zero to find critical points:**
\[
3x^2 - 3 = 0
\]
\[
x^2 = 1
\]
\[
x = \pm 1
\]

Thus, the critical points are \( x = 1 \) and \( x = -1 \).

3. **Find the second derivative:** This will help us determine whether the critical points are maxima, minima, or points of inflection.

\[
f''(x) = \frac{d}{dx}(3x^2 - 3) = 6x
\]

4. **Test the critical points using the second derivative:**

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

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

5. **Find the function values at the critical points:**

- At \( x = 1 \):
\[
f(1) = (1)^3 - 3(1) + 11 = 1 - 3 + 11 = 9
\]
So, \( (1, 9) \) is a local minimum.

- At \( x = -1 \):
\[
f(-1) = (-1)^3 - 3(-1) + 11 = -1 + 3 + 11 = 13
\]
So, \( (-1, 13) \) is a local maximum.

### Conclusion:
- The local maximum occurs at \( x = -1 \) with a value of 13.
- The local minimum occurs at \( x = 1 \) with a value of 9.

Would you like more details on any part of this solution?

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Here are 8 related questions:

1. How do you interpret the significance of critical points in a function?
2. Can you explain the relationship between the first and second derivatives and the shape of the graph?
3. How do we identify inflection points in a function?
4. What happens if the second derivative equals zero at a critical point?
5. Can a function have more than one maximum or minimum point?
6. What is the difference between absolute and local extrema?
7. How would you find the inflection points for this function?
8. What are saddle points, and how do they relate to this process?

**Tip:** To confirm the behavior of the function near critical points, sketching a graph of the function is often helpful in visualizing maxima, minima, and inflection points.

Discover how to find the maxima and minima of the function f(x) = x^3 - 3x + 11 using calculus. This detailed solution covers the calculation of critical points, determination of local extrema, and the use of derivatives in calculus. Suitable for advanced high school students.

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