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The problem asks to find the points on the graph of the function \( f(x) = 27x - x^3 \) where the tangent line is horizontal.

### Steps to solve:
1. **Condition for a horizontal tangent line**: A tangent line is horizontal where the derivative of the function is zero, i.e., \( f'(x) = 0 \).
   
2. **Find the derivative**: The derivative of \( f(x) = 27x - x^3 \) is:
   \[
   f'(x) = 27 - 3x^2
   \]

3. **Set the derivative equal to zero**: To find where the tangent line is horizontal, we solve \( f'(x) = 0 \):
   \[
   27 - 3x^2 = 0
   \]
   Simplifying:
   \[
   3x^2 = 27
   \]
   \[
   x^2 = 9
   \]
   \[
   x = \pm 3
   \]

4. **Find the corresponding points**: Now we substitute \( x = 3 \) and \( x = -3 \) into the original function \( f(x) = 27x - x^3 \) to find the points.

   - For \( x = 3 \):
     \[
     f(3) = 27(3) - 3^3 = 81 - 27 = 54
     \]
     So, one point is \( (3, 54) \).

   - For \( x = -3 \):
     \[
     f(-3) = 27(-3) - (-3)^3 = -81 + 27 = -54
     \]
     So, the second point is \( (-3, -54) \).

### Conclusion:
The points where the tangent line is horizontal are \( (3, 54) \) and \( (-3, -54) \).

Would you like further details on this solution, or do you have any questions?

Here are some related questions:
1. What is the significance of a horizontal tangent line in a graph?
2. How does the derivative relate to the slope of the tangent line?
3. What happens to the shape of the graph around the points where the tangent is horizontal?
4. How would we determine if these points are local maxima or minima?
5. Could there be any inflection points in this graph? If so, how would we find them?

**Tip:** Horizontal tangent lines typically occur at local maxima, minima, or saddle points where the slope of the curve is zero.

Find the points on the graph of f(x) = 27x - x^3 where the tangent line is horizontal.

In this calculus problem, find the points where the tangent line to the function f(x) = 27x - x^3 is horizontal by solving for the derivative and setting it to zero. This problem demonstrates key calculus concepts such as derivatives and critical points.

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