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We are tasked with finding the point(s) on the graph of the function \( f(x) = x^3 - 12x + 1 \) where the function has a horizontal tangent line.

### Step 1: Understand the problem
- A **horizontal tangent line** occurs when the slope of the tangent line is zero.
- The slope of the tangent line is given by the derivative of the function \( f(x) \).

### Step 2: Find the derivative of \( f(x) \)
We differentiate \( f(x) = x^3 - 12x + 1 \) with respect to \( x \):

\[
f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(12x) + \frac{d}{dx}(1)
\]

Using basic differentiation rules:
\[
f'(x) = 3x^2 - 12
\]

### Step 3: Set the derivative equal to zero
To find the points where the tangent line is horizontal, we need to set the derivative equal to zero and solve for \( x \):

\[
3x^2 - 12 = 0
\]

### Step 4: Solve for \( x \)
Simplify and solve the equation:

\[
3x^2 = 12
\]
\[
x^2 = 4
\]
\[
x = \pm 2
\]

So, the function has horizontal tangent lines at \( x = 2 \) and \( x = -2 \).

### Step 5: Find the corresponding \( y \)-values
To find the points, we substitute \( x = 2 \) and \( x = -2 \) into the original function \( f(x) = x^3 - 12x + 1 \):

- For \( x = 2 \):
  \[
  f(2) = 2^3 - 12(2) + 1 = 8 - 24 + 1 = -15
  \]

- For \( x = -2 \):
  \[
  f(-2) = (-2)^3 - 12(-2) + 1 = -8 + 24 + 1 = 17
  \]

### Step 6: Conclusion
The points on the graph where the function has horizontal tangent lines are:
- \( (2, -15) \)
- \( (-2, 17) \)

Would you like any additional details or clarifications?

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Here are 5 related questions to deepen your understanding:
1. What is the geometric meaning of a horizontal tangent line in terms of the graph's shape?
2. How do we determine whether the horizontal tangent line corresponds to a local maximum or minimum?
3. How does the second derivative of a function help classify critical points (such as those with horizontal tangents)?
4. Can a cubic function have more than two points with horizontal tangent lines?
5. How would the function's behavior change if we altered the constants in \( f(x) = x^3 - 12x + 1 \)?

**Tip:** When solving for critical points, always check the second derivative to classify the nature of the critical points (whether they are maxima, minima, or points of inflection).

Find the point(s) on the graph of f(x) = x^3 - 12x + 1 for which the function has a horizontal tangent line.

Learn how to find the points where the function f(x) = x^3 - 12x + 1 has a horizontal tangent line. This detailed solution involves differentiation and solving for critical points. Suitable for Grades 10-12 students.

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