To solve this problem, we need to integrate the derivative $f'(x)$ to find the original function $f(x)$. Additionally, since $f(5) = 3$ is given, we can determine the constant of integration and calculate $f(x)$ at other specific points. Here's the process:

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Part (a): Finding $f(x)$ at specific points

Step 1: Analyze $f'(x)$

The derivative $f'(x)$ is piecewise-linear based on the graph:

• From $x = 0$ to $x = 1$: $f'(x) = -1$
• From $x = 1$ to $x = 4$: $f'(x)$ increases linearly from $-1$ to $2$
• From $x = 4$ to $x = 8$: $f'(x)$ decreases linearly from $2$ to $-2$

Step 2: Integrate $f'(x)$ piecewise

The integral of $f'(x)$ gives $f(x)$. Start with $f(5) = 3$ as the anchor point and integrate piece by piece, adjusting the constant of integration as necessary.

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$f(0)$:

To determine $f(0)$, integrate $f'(x)$ over the interval $[0, 5]$, keeping track of changes in slope and adding the appropriate constants.

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Would you like me to compute $f(0)$ in detail step by step and proceed to graph the function $f(x)$ as described? Let me know! 😊

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Follow-Up Questions:

1. How do the slopes in $f'(x)$ affect the concavity of $f(x)$?
2. What does $f(5) = 3$ imply for calculating the constant of integration?
3. Why is the piecewise nature of $f'(x)$ important for $f(x)$?
4. How does the derivative graph help us sketch the original function?
5. How does continuity of $f(x)$ impact the integration?

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Tip: Always verify the consistency of integration bounds with given conditions to ensure accuracy in your solution.