If a function f has derivative f'(x) > 0 for all x in an interval, what can be said about f on that interval?

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

If a function f has derivative f'(x) > 0 for all x in an interval, what can be said about f on that interval?

Explanation:
Whenever a function has a derivative positive at every point of an interval, the function must rise as x increases. For any two points x1 < x2 in the interval, the Mean Value Theorem gives f(x2) − f(x1) = f'(c)(x2 − x1) for some c in (x1, x2). Since f'(c) > 0 and x2 − x1 > 0, the difference is positive, so f(x2) > f(x1). Therefore, f is strictly increasing on the interval. This also means it cannot be decreasing, constant, or non-monotonic.

Whenever a function has a derivative positive at every point of an interval, the function must rise as x increases. For any two points x1 < x2 in the interval, the Mean Value Theorem gives f(x2) − f(x1) = f'(c)(x2 − x1) for some c in (x1, x2). Since f'(c) > 0 and x2 − x1 > 0, the difference is positive, so f(x2) > f(x1). Therefore, f is strictly increasing on the interval. This also means it cannot be decreasing, constant, or non-monotonic.

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