If f'(x) > 0 on (a,b), what can be concluded about f on (a,b)?

If f'(x) > 0 on (a,b), what can be concluded about f on (a,b)?

• A. No information
• B. f is decreasing
• C. f is increasing on (a,b)
• D. f is constant

Answer: (C)

A positive derivative on the interval means the function’s slope is always upward there. For any two points x1 < x2 in (a,b), the change in the function value is f(x2) − f(x1). By the Mean Value Theorem, there exists c between x1 and x2 with f(x2) − f(x1) = f′(c)(x2 − x1). Since f′(c) > 0 and x2 − x1 > 0, their product is positive, so f(x2) > f(x1). That makes f strictly increasing on (a,b). This rules out being decreasing or constant, and there is definite information about the function’s behavior on that interval.

A positive derivative on the interval means the function’s slope is always upward there. For any two points x1 < x2 in (a,b), the change in the function value is f(x2) − f(x1). By the Mean Value Theorem, there exists c between x1 and x2 with f(x2) − f(x1) = f′(c)(x2 − x1). Since f′(c) > 0 and x2 − x1 > 0, their product is positive, so f(x2) > f(x1). That makes f strictly increasing on (a,b). This rules out being decreasing or constant, and there is definite information about the function’s behavior on that interval.