To solve this, we need to analyze the function graph and sketch the function’s derivative based on its slope changes.

Observations about the function graph:

1. Segment 1: Starts with a positive slope, increasing from left to right.
2. Segment 2: The slope becomes zero at the peak (maximum point) of the graph.
3. Segment 3: The slope turns negative as it descends after the peak.

Derivative sketch:

• For Segment 1: The derivative is positive since the function is increasing. The slope is positive, so the derivative will lie above the x-axis.
• At the peak (Segment 2): The derivative is zero, as the slope is zero.
• For Segment 3: The function is decreasing, so the derivative is negative, indicating that the derivative graph will lie below the x-axis after this point.

The rough sketch of the derivative would show:

1. A positive value in Segment 1 (above the x-axis).
2. Zero value at the peak.
3. A negative value in Segment 3 (below the x-axis).

Would you like me to create a detailed visual sketch of the derivative for this function?

Related Questions

1. How does the derivative behave at inflection points?
2. What does a negative derivative indicate about a function’s behavior?
3. How can we determine concavity from the second derivative?
4. What does it mean when the derivative of a function is zero?
5. How can critical points help in sketching a derivative?

Tip:

For a better understanding of function behavior, study how the slope (derivative) changes over each segment of the function’s graph.