To determine the correct description of the graph of the function $f(x)$, we need to analyze the function’s behavior (e.g., increasing, decreasing) by:

1. Looking at its derivative $f'(x)$, if given.
2. Visualizing the behavior of the function, which depends on its critical points and intervals of increase/decrease.

General Steps to Determine:

• If $f'(x) > 0$ on an interval, the function is increasing.
• If $f'(x) < 0$ on an interval, the function is decreasing.
• A change in the sign of $f'(x)$ (from positive to negative or vice versa) indicates a change from increasing to decreasing or vice versa.

Analyzing Choices:

• (A): The function increases throughout.
• (B): The function decreases throughout.
• (C): The function increases first and then decreases (e.g., a hill shape).
• (D): The function decreases first and then increases (e.g., a valley shape).

Please describe the graph or share the equation/details so I can provide a more specific explanation of its behavior.

Would you like detailed steps on how to determine this using calculus or an explanation of critical points? Let me know!

Related Questions:

1. What are critical points, and how do they help in determining the behavior of a graph?
2. How does the second derivative $f''(x)$ provide insight into concavity?
3. What is the role of inflection points in the graph of a function?
4. How can you sketch a function just using $f'(x)$ and $f''(x)$?
5. Why do maxima and minima occur at critical points where $f'(x) = 0$?

Tip:

Always examine both $f'(x)$ and $f''(x)$ for a complete understanding of the graph's behavior.