Insert the words “greater than,” “less than,” or “cannot be determined” into
each of the following
(a) If a function f is concave down on its entire domain of (−∞, ∞), then its average rate
of change on the interval [1, 2] is __________________________ its average rate of
change on the interval [3, 4].
(b) If a function g has a negative rate of change on (−∞, 3) and a positive rate of
change on (3, ∞), then g(3) is __________________________ g(4)

Here are the answers with explanations:

### (a)
If a function \( f \) is concave down on its entire domain, this means the slope of the tangent lines (or the rate of change) is decreasing as you move from left to right. Even though the slope is decreasing, we cannot directly compare the average rate of change on different intervals because we don't know the exact values or behavior of the function on these intervals. The concavity alone isn't sufficient to determine the comparison of average rates of change between two intervals.

Answer: **cannot be determined**

### (b)
Given that the function \( g \) has a negative rate of change on \( (-\infty, 3) \), and a positive rate of change on \( (3, \infty) \), this indicates that \( g(3) \) is likely a minimum point because the slope changes from negative to positive. Since \( g(3) \) is a minimum, we can infer that \( g(3) \) is less than \( g(4) \), as \( g(4) \) is on the increasing side of the function.

Answer: **less than**

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Do you want further details or have any questions about these comparisons?

Here are 5 related questions to expand on this concept:
1. What is the significance of concavity in determining the behavior of a function's derivative?
2. How is the average rate of change different from the instantaneous rate of change?
3. What conditions determine if a point is a local minimum or maximum in a function?
4. Can a function be concave down but still have an increasing average rate of change?
5. What is the First Derivative Test, and how does it relate to changes in function values?

**Tip:** The average rate of change of a function between two points is equivalent to the slope of the secant line connecting those points.

Insert the words “greater than,” “less than,” or “cannot be determined” into each of the following: (a) If a function f is concave down on its entire domain of (−∞, ∞), then its average rate of change on the interval [1, 2] is __________________________ its average rate of change on the interval [3, 4]. (b) If a function g has a negative rate of change on (−∞, 3) and a positive rate of change on (3, ∞), then g(3) is __________________________ g(4).

Explore how concavity affects the average rate of change of a function across different intervals and how the First Derivative Test helps identify critical points. This problem applies calculus concepts such as concavity, average rate of change, and derivatives. Suitable for college-level calculus students.

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