The problem asks to sketch the graph of a function $f(x)$ based on the following conditions:

1. $f(0) = 0$
2. $f'(0) = 3$
3. $f'(1) = 0$
4. $f'(2) = -1$

These conditions give important information about the function and its derivatives at specific points:

• $f(0) = 0$: The graph passes through the origin at $(0, 0)$.
• $f'(0) = 3$: The slope of the tangent at $x = 0$ is 3, meaning the graph is rising steeply at $x = 0$.
• $f'(1) = 0$: The slope of the tangent at $x = 1$ is 0, indicating a horizontal tangent at $x = 1$, which suggests a local maximum or minimum.
• $f'(2) = -1$: The slope at $x = 2$ is negative, meaning the graph is decreasing.

Evaluating the given options in the image:

1. First graph (top-left):

   • At $x = 0$, the graph passes through the origin, and the slope appears steep and positive, which matches $f(0) = 0$ and $f'(0) = 3$.
   • At $x = 1$, there is a horizontal tangent, which satisfies $f'(1) = 0$.
   • After $x = 1$, the graph starts decreasing, consistent with $f'(2) = -1$.

2. Second graph (top-right):

   • At $x = 0$, the graph passes through the origin, but the slope seems less steep than 3.
   • At $x = 1$, the slope is not clearly horizontal, which does not satisfy $f'(1) = 0$.

3. Third graph (bottom-left):

   • At $x = 0$, the graph does not pass through the origin, so it doesn't satisfy $f(0) = 0$.

4. Fourth graph (bottom-right):

   • The graph seems to meet some conditions, but at $x = 0$, the slope appears incorrect for $f'(0) = 3$, and the graph doesn't match all conditions.

Conclusion:

The first graph (top-left) is the correct choice. It satisfies all the given conditions for $f(x)$, including the correct values of the function and its derivatives at the specified points.

Would you like more details, or do you have any questions about this?

Here are 5 related questions for further exploration:

1. What is the significance of a function's derivative being zero at a point?
2. How can you determine concavity from a graph?
3. What does it mean when the derivative of a function is positive or negative?
4. How can you find local maxima and minima using the first derivative?
5. What is the geometric interpretation of the second derivative?

Tip: A function’s first derivative gives the slope of the tangent line, while the second derivative provides information about the curvature (concavity) of the graph.