If a function is differentiable on (a,b) and increasing on that interval, what can be said about f'(x) on (a,b)?

• A. f'(x) ≤ 0 for all x in (a,b)
• B. f'(x) = 0 for all x in (a,b)
• C. f'(x) > 0 for all x in (a,b)
• D. f'(x) ≥ 0 for all x in (a,b)

Answer: (D)

The main idea is that an increasing function cannot have a negative instantaneous rate of change anywhere in its domain where the derivative exists. For any point x in (a,b), look at small steps forward and backward. If you move a little to the right (h > 0 with x + h in (a,b)), the function value can only stay the same or increase, so the difference quotient [f(x+h) − f(x)]/h ≥ 0. If you move a little to the left (h < 0 with x + h in (a,b)), the function value can only stay the same or decrease, but dividing by a negative h flips the sign, so the quotient [f(x+h) − f(x)]/h ≥ 0 as well. Since the derivative at x is the limit of these quotients as h → 0, that limit must be ≥ 0. So f′(x) ≥ 0 for every x in (a,b). The derivative may be zero at some points and positive at others, but it cannot be negative.