To find $H'(1)$ given that $H(x) = 4f(x) + g(x)$, we need to differentiate $H(x)$:

$H'(x) = 4f'(x) + g'(x)$

We then substitute $x = 1$:

$H'(1) = 4f'(1) + g'(1)$

Now, we need to find the derivatives $f'(1)$ and $g'(1)$ using the graphs of $f(x)$ and $g(x)$.

Step 1: Find $f'(1)$

From the graph, the function $f(x)$ is a straight line (blue line), so its slope is constant. The slope can be found as:

$f'(x) = \frac{\Delta y}{\Delta x} = \frac{4 - 0}{4 - 0} = 1$

Thus, $f'(1) = 1$.

Step 2: Find $g'(1)$

Similarly, for $g(x)$ (red line), its slope is also constant:

$g'(x) = \frac{\Delta y}{\Delta x} = \frac{0 - 10}{4 - 0} = -\frac{10}{4} = -2.5$

Thus, $g'(1) = -2.5$.

Step 3: Substitute into the derivative of $H(x)$

$H'(1) = 4 \cdot 1 + (-2.5) = 4 - 2.5 = 1.5$

Final Answer:

$H'(1) = 1.5$

Would you like further details or have any questions? Here are some related questions to explore:

1. How do you differentiate a linear combination of functions?
2. How do you find the slope of a line from a graph?
3. What is the significance of the derivative in understanding the rate of change?
4. Can we find the second derivative $H''(x)$ using the given functions?
5. What are other real-life applications of derivatives in problem-solving?

Tip: When dealing with graphs of functions, always look for consistent patterns in slope to simplify derivative calculations!