Understanding Function Rates of Change and Concavity

Published on June 6, 2026

Calculus

Analyze the relationship between the first and second derivatives of a function to understand whether it's increasing/decreasing at an increasing/decreasing rate and determine the concavity of its graph.

Question

Are these graph descriptions correct ? What would be the second derivatives of each of these ?

Answer

Based on the image provided, we can analyze the descriptions of the graphs and determine the signs of their first and second derivatives.

Analysis of Graph Descriptions

The text defines the relationship between the rate of change (first derivative) and the concavity (second derivative). Let’s verify each one:

Increasing at an increasing rate:
- Description: is increasing and concave up.
- Verification: Correct. If a function is increasing, its slope is positive. If it is increasing at an increasing rate, the slope is becoming more positive (steeper), which means the second derivative is positive (concave up).
Increasing at a decreasing rate:
- Description: is increasing and concave down.
- Verification: Correct. The function is still going up (positive slope), but the slope is getting smaller (less steep), which means the second derivative is negative (concave down).
Decreasing at an increasing rate:
- Description: is decreasing and concave down.
- Verification: Correct. “Increasing rate” in the context of a decreasing function usually refers to the magnitude of the rate of change. As the function drops faster and faster, the slope becomes more negative (e.g., from to ). A decreasing slope means the second derivative is negative (concave down).
Decreasing at a decreasing rate:
- Description: is decreasing and concave up.
- Verification: Correct. The function is going down, but it is leveling off. The slope is becoming less negative (e.g., from to ), meaning the slope is increasing. Thus, the second derivative is positive (concave up).

Conclusion: Yes, the graph descriptions in the image are correct.

Second Derivatives of Each Case

The second derivative

determines the concavity of the function. Based on the analysis above, here are the signs of the first and second derivatives for each case:

Description	First Derivative	Second Derivative	Concavity
Increasing at an increasing rate	Positive ()	Positive ()	Concave Up
Increasing at a decreasing rate	Positive ()	Negative ()	Concave Down
Decreasing at an increasing rate	Negative ()	Negative ()	Concave Down
Decreasing at a decreasing rate	Negative ()	Positive ()	Concave Up

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