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?

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.