Let's analyze the problem inside the green rectangle:
We are given the function:

$f(x) = x^2 - 6x - 8 \ln x$

The task is: To determine for which values of $x$ the derivative of the function $f(x)$ is equal to zero.
That is, solve for $x$ such that $f'(x) = 0$.

Step-by-Step Plan

Step 1: Find the Derivative $f'(x)$

• Take the derivative of each term:
  • The derivative of $x^2$ is $2x$.
  • The derivative of $-6x$ is $-6$.
  • The derivative of $-8 \ln x$ is $-8 \cdot \frac{1}{x}$.

Step 2: Write Down the Expression for $f'(x)$

• Combine all derivatives to get one expression.

Step 3: Set $f'(x)$ Equal to Zero and Solve

• Set the entire derivative expression to zero, and solve for $x$.
• Show every step of simplification.

Step 4: Verify Logarithmic Domain

• Remember that $\ln x$ is defined only for $x > 0$.

Let's proceed step by step.

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Step 1: Find the derivative of $f(x)$.

$f(x) = x^2 - 6x - 8 \ln x$

$f'(x) = \frac{d}{dx} (x^2) - \frac{d}{dx} (6x) - \frac{d}{dx} (8 \ln x)$

The derivative is:

$f'(x) = 2x - 6 - \frac{8}{x}$

Step 2: Set the derivative equal to zero and write the equation:

$2x - 6 - \frac{8}{x} = 0$

Step 3: Solve for $x$

Let's solve this equation step by step.

First, move $-\frac{8}{x}$ to the other side:

$$2x - 6 = \frac{8}{x}$$

Now, multiply both sides by $x$ to clear the fraction ($x > 0$ because of $\ln x$):

$$x(2x - 6) = 8$$ $$2x^2 - 6x = 8$$

Next, subtract 8 from both sides:

$$2x^2 - 6x - 8 = 0$$

Now, let's solve this quadratic equation for $x$:

$$2x^2 - 6x - 8 = 0$$

Let's find the values of $x$. The quadratic equation gives us two solutions:

$x = -1 \quad \text{and} \quad x = 4$

Step 4: Check the domain: Recall that $f(x)$ includes $\ln x$, so $x > 0$.

• $x = -1$ is not allowed.
• $x = 4$ is allowed.

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Final Answer:
The value of $x$ for which the derivative of the function $f(x) = x^2 - 6x - 8 \ln x$ is zero:

$$\boxed{x = 4}$$