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show work for each step including basic differentiation

To find the points on the graph of \( f(x) = x^3 - 12x + 1 \) where the function has a horizontal tangent line, we need to find where the derivative of the function, \( f'(x) \), is equal to zero.

### Step 1: Differentiate the function
The function given is:
\[ f(x) = x^3 - 12x + 1 \]

We apply the power rule for differentiation:
\[ f'(x) = 3x^2 - 12 \]

### Step 2: Set the derivative equal to zero to find critical points
\[ 3x^2 - 12 = 0 \]

### Step 3: Solve the equation
\[ 3x^2 = 12 \]
\[ x^2 = 4 \]
\[ x = \pm 2 \]

### Step 4: Calculate the y-coordinates of these points
Substitute \( x = 2 \) and \( x = -2 \) back into the original function \( f(x) \):
\[ f(2) = 2^3 - 12 \cdot 2 + 1 = 8 - 24 + 1 = -15 \]
\[ f(-2) = (-2)^3 - 12 \cdot (-2) + 1 = -8 + 24 + 1 = 17 \]

### Points with horizontal tangent lines:
The points on the graph where the function has a horizontal tangent line are:
\[ (2, -15) \text{ and } (-2, 17) \]

These points are where the slope of the tangent (given by the derivative) is zero, indicating horizontal tangents.

If you have further questions or need more details on any step, feel free to ask. Here are a few related questions you might find interesting:

1. What does the derivative tell us about the function beyond just the slope of tangents?
2. How would the process change if we were looking for vertical tangents instead?
3. What are the implications of finding multiple points with horizontal tangents on the graph's shape?
4. How can we determine the concavity of the function at these points?
5. What would the graph of this function look like around these points?

**Tip:** When solving for where derivatives equal zero, always consider the possibility of multiple solutions, as critical points can indicate changes in the increasing/decreasing nature of the function or the presence of local maxima or minima.

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. This solution includes step-by-step differentiation and solving for critical points. Suitable for high school students in Grades 10-12.

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