We are given that $G(x) = 4f(x) - g(x)$, where $f(x)$ and $g(x)$ are shown in the graph, and we need to find $G'(1)$, which is the derivative of $G(x)$ at $x = 1$.

Step-by-step solution:

1. Derivative of G(x): $G'(x) = \frac{d}{dx} [4f(x) - g(x)] = 4f'(x) - g'(x)$ So, to find $G'(1)$, we need to calculate $f'(1)$ and $g'(1)$, the derivatives of $f(x)$ and $g(x)$ at $x = 1$.

2. Determine slopes from the graph: The derivative at a point is the slope of the tangent line at that point. From the graph, we can estimate the slopes of $f(x)$ and $g(x)$ at $x = 1$.

   • The function $f(x)$ appears to be a straight line with a positive slope in the segment where $x = 1$. We can calculate the slope by observing the rise over run between two points on this line.

     For $f(x)$:

     • Between $(0, 2)$ and $(2, 4)$, the slope is: $f'(x) = \frac{4 - 2}{2 - 0} = \frac{2}{2} = 1$ So, $f'(1) = 1$.

   • The function $g(x)$ also appears linear in the segment where $x = 1$. From the graph, we observe:

     For $g(x)$:

     • Between $(0, 0)$ and $(2, 4)$, the slope is: $g'(x) = \frac{4 - 0}{2 - 0} = \frac{4}{2} = 2$ So, $g'(1) = 2$.

3. Calculate $G'(1)$: Now that we have $f'(1) = 1$ and $g'(1) = 2$, we can substitute into the expression for $G'(x)$: $G'(1) = 4f'(1) - g'(1) = 4(1) - 2 = 4 - 2 = 2$

Final Answer:

$G'(1) = 2$

Would you like further details on any step, or have any questions about this solution?

Here are five related questions to explore:

1. How do you find the derivative of a function from a graph if the function is not linear?
2. What is the geometric interpretation of the derivative at a point?
3. How do we differentiate a product of functions?
4. Can the second derivative give us more information about the graph's behavior at $x = 1$?
5. How do you compute the derivative of a composite function?

Tip: When interpreting graphs, remember that the steeper the line, the larger the magnitude of the derivative.