Photo AI
The graph of a cubic function $f$ is shown on the right - Leaving Cert Mathematics - Question 6 - 2012
Question 6
The graph of a cubic function $f$ is shown on the right. One of the four diagrams A, B, C, D below shows the graph of the derivative of $f$. State which one it is,... show full transcript
Worked Solution & Example Answer:The graph of a cubic function $f$ is shown on the right - Leaving Cert Mathematics - Question 6 - 2012
Step 1
Step 2
Justification
Answer
To justify this choice, we consider the behavior of the cubic function . The derivative of a function represents the slope of the tangent line at any given point.
-
Analyzing the Tangent Lines:
- In the regions where the cubic function is increasing, the slope is positive, leading the derivative to be positive.
- Where the function decreases, the slope is negative, meaning the derivative is negative.
- Graph C shows a downward sloping parabola, indicating that it starts positive, moves negative, and never reaches a horizontal line, thus correctly reflects the nature of the cubic function.
-
Excluding Other Graphs:
- Graph A is incorrect as it reaches a horizontal point, indicating a derivative of zero.
- Graph B and Graph D also show slopes that are impossible for this cubic function as they consistently cross the x-axis.
-
Conclusion:
- Given the analysis, graph C accurately conveys the relationship between the cubic function and its derivative.