The problem requires finding the derivative of the function $g(x) = 6f(x) - 3x$ at $x = x_0$, where $f(x)$ is the function graphed, and the tangent line to $y = f(x)$ at $x_0$ is shown.

Step-by-Step Solution:

1. Identify the derivative of $g(x)$: The derivative of $g(x) = 6f(x) - 3x$ is: $g'(x) = 6f'(x) - 3$ Here, $f'(x)$ is the slope of the tangent line to $f(x)$ at $x = x_0$.

2. Determine $f'(x_0)$: The slope of the tangent line is visually represented in the graph. The tangent line passes through the points:

   • At $x = 0$, $y = 1$ (point $O$).
   • At $x = 1$, $y = 0$.

   The slope of the tangent line is given by: $f'(x_0) = \frac{\Delta y}{\Delta x} = \frac{0 - 1}{1 - 0} = -1$

3. Substitute $f'(x_0) = -1$ into $g'(x)$: Now, substituting into $g'(x) = 6f'(x) - 3$: $g'(x_0) = 6(-1) - 3 = -6 - 3 = -9$

Final Answer:

The value of $g'(x_0)$ is -9.

Would you like further clarification on any step or additional examples of similar problems? Let me know!

Related Questions:

1. What is the derivative of a linear combination of two functions?
2. How do you find the slope of a tangent line from a graph?
3. How does a tangent line help approximate derivatives?
4. Can you explain the derivative rules for scalar multiplication and subtraction?
5. What happens to $g(x) = 6f(x) - 3x$ when $f'(x)$ is zero?

Tip:

Always identify the slope of the tangent line carefully, as it represents the derivative of the function at the point of tangency.