Which of the following best describes the difference between a relative maximum and an absolute maximum of a function on an interval $I$?

• A. A relative maximum is always greater than the absolute maximum on the interval $I$.
• B. An absolute maximum only occurs at the endpoints of the interval, while a relative maximum can occur anywhere.
• C. There is no difference; both terms refer to the highest point on the interval $I$.
• D. A relative maximum is a point where the function value is greater than or equal to all nearby values, while an absolute maximum is a point where the function value is greater than or equal to all values on the entire interval $I$.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of a relative maximum. A relative maximum is a point on the graph of a function where the function value is greater than or equal to all nearby values within a small neighborhood around that point. This does not necessarily mean it is the highest value on the entire interval.
2. Step 2: Understand the concept of an absolute maximum. An absolute maximum is a point on the graph of a function where the function value is greater than or equal to all values of the function on the entire interval. This is the highest point on the interval.
3. Step 3: Compare the two concepts. A relative maximum is localized and depends on nearby values, while an absolute maximum is global and considers the entire interval.
4. Step 4: Note that a relative maximum can occur anywhere within the interval, including endpoints or interior points, while an absolute maximum can occur at endpoints or interior points depending on the function's behavior.
5. Step 5: Conclude that the correct answer is: A relative maximum is a point where the function value is greater than or equal to all nearby values, while an absolute maximum is a point where the function value is greater than or equal to all values on the entire interval.