If f(x) is continuous on [a, b], which of the following must be true?

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Question: If f(x) is continuous on [a, b], which of the following must be true?

Options:

Correct Answer: f(x) takes every value between f(a) and f(b)

Solution:

By the Intermediate Value Theorem, a continuous function on a closed interval takes every value between f(a) and f(b).

If f(x) is continuous on [a, b], which of the following must be true?

If f(x) is continuous on [a, b], which of the following must be true?

Step 1: Understand that f(x) is a function that is continuous on the interval from a to b.
Step 2: Recognize that 'continuous' means there are no breaks, jumps, or holes in the graph of the function between a and b.
Step 3: Identify the values of the function at the endpoints: f(a) is the value at the start (a) and f(b) is the value at the end (b).
Step 4: The Intermediate Value Theorem states that if you have a continuous function, it will take every value between f(a) and f(b).
Step 5: Conclude that for any value y between f(a) and f(b), there exists at least one x in the interval [a, b] such that f(x) = y.

Intermediate Value Theorem – A theorem stating that if a function is continuous on a closed interval, it takes every value between its endpoints.