https://file.aiquickdraw.com/mathbot/file/1729007036264rwuzpo64.jpg

Let's analyze the given table and determine which description best fits the behavior of the function \( f(x) \).

### Table of values:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 100 \\
4 & 58 \\
8 & 32 \\
12 & 24 \\
16 & 20 \\
20 & 18 \\
24 & 17 \\
\hline
\end{array}
\]

### Step-by-step analysis:

1. **Is \( f(x) \) increasing or decreasing?**
   - As \( x \) increases from 0 to 24, the values of \( f(x) \) are decreasing. For instance:
     \[
     f(0) = 100, f(4) = 58, f(8) = 32, f(12) = 24, f(16) = 20, f(20) = 18, f(24) = 17
     \]
   - The function is consistently **decreasing**.

2. **Is the function concave up or concave down?**
   - To determine concavity, we need to consider the rate of change of \( f(x) \), i.e., the second derivative.
   - Look at how the differences between consecutive \( f(x) \) values change:
     \[
     f(0) - f(4) = 42, \quad f(4) - f(8) = 26, \quad f(8) - f(12) = 8, \quad f(12) - f(16) = 4, \quad f(16) - f(20) = 2, \quad f(20) - f(24) = 1
     \]
   - The differences are **decreasing**, which suggests that the rate of decrease is slowing down, indicating that the function is **concave up**.

### Conclusion:
The function \( f(x) \) is **decreasing** and **concave up**.

The correct option is:
- \( f(x) \) is decreasing and concave up.

Would you like more details or have any questions? Here are 5 follow-up questions you may find useful:

1. How do we formally determine concavity using the second derivative?
2. What does it mean for a function to be concave up?
3. Can a function be both increasing and concave up?
4. How would the description change if the function were increasing?
5. What are some real-world examples of concave up functions?

**Tip:** To check concavity from data, look at how the changes between values evolve. If the changes get smaller in magnitude, it suggests concave up behavior.

Below are the values of a function f(x). Which of the following best describes the graph of f(x)?

Analyze the given function values to determine whether f(x) is increasing or decreasing and concave up or down. This problem involves recognizing decreasing functions and determining concavity using the rate of change between consecutive values. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation