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The image contains the following mathematical problem:

1. A function \( f_k(x) = x^2(x - k) \), where \( k \) is a variable.
2. A point \( P = (2, ?) \), where the \( y \)-coordinate is not provided yet.
3. The task is to find the **general tangent equation** at the point \( P \) with respect to the variable \( k \).

Let's solve this step-by-step:

### Step 1: Find the \( y \)-coordinate of point \( P \)
Given that \( P = (2, ?) \), we can plug \( x = 2 \) into the function \( f_k(x) = x^2(x - k) \) to find the \( y \)-coordinate.

\[
f_k(2) = 2^2(2 - k) = 4(2 - k) = 8 - 4k
\]
So, \( P = (2, 8 - 4k) \).

### Step 2: Find the derivative of \( f_k(x) \)
The general formula for the tangent line requires us to find the slope at point \( P \), which is the derivative of \( f_k(x) \) evaluated at \( x = 2 \).

\[
f_k(x) = x^2(x - k)
\]

Applying the product rule to find \( f_k'(x) \):

\[
f_k'(x) = \frac{d}{dx}[x^2] \cdot (x - k) + x^2 \cdot \frac{d}{dx}(x - k)
\]
\[
f_k'(x) = 2x(x - k) + x^2(1) = 2x(x - k) + x^2
\]

Now, evaluate this at \( x = 2 \):

\[
f_k'(2) = 2(2)(2 - k) + 2^2 = 4(2 - k) + 4 = 8 - 4k + 4 = 12 - 4k
\]

### Step 3: Write the equation of the tangent line
The general equation of the tangent line at a point \( (x_1, y_1) \) is:

\[
y - y_1 = m(x - x_1)
\]

Here:
- \( m = 12 - 4k \) (the slope),
- \( x_1 = 2 \),
- \( y_1 = 8 - 4k \).

Substituting these values into the tangent line equation:

\[
y - (8 - 4k) = (12 - 4k)(x - 2)
\]

This is the general tangent line equation in terms of \( k \) at the point \( P(2, 8 - 4k) \).

Would you like more details or further clarifications?

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Here are five related questions for deeper understanding:

1. What happens to the tangent line if \( k = 2 \)?
2. How does the slope of the tangent line change as \( k \) increases?
3. How do we interpret the meaning of \( k \) geometrically?
4. Can you derive the second derivative of \( f_k(x) \) and discuss its concavity?
5. How would you find the intersection points of the tangent line with the axes?

**Tip:** The product rule is often essential when differentiating functions that are products of two or more terms.

Find the general tangent equation for the function f_k(x) = x^2(x - k) at point P = (2, ?) with respect to the variable k.

Learn how to find the general tangent formula for the function f_k(x) = x^2(x - k) at the point P = (2, ?), including the use of the product rule for derivatives and applying the tangent line equation. This problem is suitable for students in Grades 11-12 or first-year university.

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