from matplotlib import pyplot as plt import numpy as np

Define the derivative f'(x) from the graph

x = np.linspace(0, 8, 500) f_prime = np.piecewise(x, [x < 3, (x >= 3) & (x < 4), (x >= 4) & (x < 6), x >= 6], [lambda x: 2np.sin(np.pix/3), lambda x: -2*(x-3), lambda x: 2*(x-5), lambda x: -3*(x-6)])

Compute f(x) using integration

f = np.cumsum(f_prime) * (x[1] - x[0]) # Numerical integration f = f - f[0] # Adjust to make f(0) = 0

Plot f(x)

plt.figure(figsize=(8, 6)) plt.plot(x, f, label="$f(x)$", color="blue") plt.axhline(0, color="black", linestyle="--", linewidth=0.7) plt.axvline(0, color="black", linestyle="--", linewidth=0.7) plt.title("Sketch of $f(x)$ based on $f'(x)$") plt.xlabel("$x$") plt.ylabel("$f(x)$") plt.legend() plt.grid(alpha=0.3) plt.show() <<ImageDisplayed>>successHere is a possible sketch of the continuous function $f(x)$, based on the given derivative $f'(x)$ and the condition $f(0) = 0$. The shape of $f(x)$ reflects the behavior of $f'(x)$:

• $f(x)$ increases where $f'(x) > 0$.
• $f(x)$ decreases where $f'(x) < 0$.
• $f(x)$ has local maxima or minima where $f'(x) = 0$ and changes sign.

Would you like to analyze specific points, discuss critical points, or explore how this curve behaves in more detail?

Here are 5 related questions for exploration:

1. What are the critical points of $f(x)$, and how do they relate to $f'(x)$?
2. How can the concavity of $f(x)$ be determined using $f'(x)$?
3. Could you provide a numerical calculation of $f(x)$ at specific values of $x$?
4. How does the behavior of $f'(x)$ at $x = 6$ influence $f(x)$?
5. What happens to $f(x)$ as $x$ approaches 8, based on $f'(x)$?

Tip: Use the relationship $f(x) = \int f'(x) \, dx$ to reconstruct $f(x)$ if you are given the derivative.