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To find the point on the graph of \( f(x) = \frac{1}{3}x^3 - 3x^2 + 9x - 9 \) where the tangent line is horizontal, we need to follow these steps:

1. **Find the derivative of the function** \( f(x) \), as the derivative gives the slope of the tangent line at any point.
2. **Set the derivative equal to zero**, because the slope of a horizontal line is 0.
3. **Solve for \( x \)**, and then substitute the values of \( x \) back into the original function to find the corresponding \( y \)-coordinates.

### Step 1: Differentiate the function
The derivative of \( f(x) = \frac{1}{3}x^3 - 3x^2 + 9x - 9 \) is:

\[
f'(x) = x^2 - 6x + 9
\]

### Step 2: Set the derivative equal to zero
To find where the tangent is horizontal, set \( f'(x) = 0 \):

\[
x^2 - 6x + 9 = 0
\]

This is a quadratic equation. We can solve it by factoring:

\[
(x - 3)(x - 3) = 0
\]

Thus, \( x = 3 \).

### Step 3: Find the corresponding \( y \)-coordinate
Substitute \( x = 3 \) into the original function \( f(x) \):

\[
f(3) = \frac{1}{3}(3)^3 - 3(3)^2 + 9(3) - 9
\]
\[
f(3) = \frac{1}{3}(27) - 3(9) + 27 - 9
\]
\[
f(3) = 9 - 27 + 27 - 9 = 0
\]

So, the point where the tangent line is horizontal is \( (3, 0) \).

### Final Answer:
The point on the graph where the tangent line is horizontal is \( (3, 0) \).

Would you like more details on any of the steps or any additional explanations?

Here are 5 related questions you could explore:
1. What is the significance of the second derivative at this point?
2. Can a cubic function have more than one point where the tangent is horizontal?
3. How can we determine whether the point \( (3, 0) \) is a local minimum or maximum?
4. What happens to the behavior of the graph around \( x = 3 \)?
5. How would we find the equation of the tangent line at \( x = 3 \) if it were not horizontal?

**Tip:** Horizontal tangents often occur at critical points, which could correspond to local maxima, minima, or points of inflection.

Find the point on the graph of f(x) = 1/3 x^3 - 3x^2 + 9x - 9 at which the tangent line is horizontal.

Learn how to find the point on the graph of f(x) = 1/3x^3 - 3x^2 + 9x - 9 where the tangent line is horizontal. This involves finding the derivative and solving for when the slope is zero. Suitable for high school calculus students.

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