If f is increasing on an interval, what can be said about f' on that interval?

If f is increasing on an interval, what can be said about f' on that interval?

• A. f'(x) > 0 on the interval
• B. f'(x) < 0 on the interval
• C. f'(x) = 0 on the interval
• D. f'(x) changes sign on the interval

Answer: (A)

The derivative represents the slope of the tangent, i.e., the instantaneous rate of change. If f is increasing as x goes to the right, then small steps to the right raise the function value, so the tangent slope must be positive. Therefore, at every point where the derivative exists on that interval, f′(x) is positive. In many contexts this is taken as f′(x) > 0 across the interval, reflecting the positive rate of change as you move to the right. (If the function is strictly increasing, the average rate of change is positive for small right steps, and its limit—the derivative—will be nonnegative; it can be zero at isolated points in some cases, but the overall behavior is a positive slope.)

The derivative represents the slope of the tangent, i.e., the instantaneous rate of change. If f is increasing as x goes to the right, then small steps to the right raise the function value, so the tangent slope must be positive. Therefore, at every point where the derivative exists on that interval, f′(x) is positive. In many contexts this is taken as f′(x) > 0 across the interval, reflecting the positive rate of change as you move to the right. (If the function is strictly increasing, the average rate of change is positive for small right steps, and its limit—the derivative—will be nonnegative; it can be zero at isolated points in some cases, but the overall behavior is a positive slope.)