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We are given the following information:

- \( f(1) = 6 \)
- \( f'(1) = -6 \)
- \( g(x) = 3x f(x) \)

We are asked to write the equation of the line tangent to the graph of \( g(x) \) at \( x = 1 \).

### Step 1: Find \( g(1) \)
First, we need to calculate \( g(1) \).

\[
g(x) = 3x f(x)
\]
Substituting \( x = 1 \) and using \( f(1) = 6 \):
\[
g(1) = 3(1)f(1) = 3 \cdot 6 = 18
\]

### Step 2: Find \( g'(x) \)
Now, let's differentiate \( g(x) \) using the product rule. The product rule states:
\[
(g(x))' = 3 \cdot f(x) + 3x \cdot f'(x)
\]
Thus,
\[
g'(x) = 3f(x) + 3x f'(x)
\]

### Step 3: Find \( g'(1) \)
Substitute \( x = 1 \), \( f(1) = 6 \), and \( f'(1) = -6 \) into the expression for \( g'(x) \):

\[
g'(1) = 3f(1) + 3(1)f'(1) = 3(6) + 3(1)(-6) = 18 - 18 = 0
\]

### Step 4: Write the equation of the tangent line
The general form of the equation of a tangent line is:
\[
y - g(1) = g'(1)(x - 1)
\]

Substitute \( g(1) = 18 \) and \( g'(1) = 0 \) into this equation:
\[
y - 18 = 0(x - 1)
\]

Thus, the equation of the tangent line simplifies to:
\[
y = 18
\]

### Final Answer:
The equation of the tangent line to the graph of \( g(x) \) at \( x = 1 \) is \( y = 18 \).

---

Do you have any questions or would you like a detailed explanation on any part? 

Here are 5 related questions to consider:
1. How is the product rule used when differentiating products of functions?
2. How would the equation of the tangent line change if \( g'(1) \) were not zero?
3. Can the same method be applied if \( g(x) \) were more complex (e.g., involving trigonometric functions)?
4. How can second derivatives help in finding curvature at a point?
5. What is the significance of the slope of the tangent line being zero?

**Tip:** When working with tangent lines, always ensure you have both the function value and its derivative at the point of interest to properly form the equation.

Let f be a differentiable function with f(1) = 6 and f'(1) = -6. Let the function g(x) = 3x f(x). Write the equation of the line tangent to the graph of g at the point where x = 1.

Learn how to find the tangent line to the function g(x) = 3x f(x) at the point where x = 1. This solution uses the product rule and tangent line formula, applying key calculus concepts. Suitable for Grades 11-12.

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