Which conditions must hold for the Mean Value Theorem to apply?

Which conditions must hold for the Mean Value Theorem to apply?

• A. f continuous on (a,b) and differentiable on [a,b]
• B. f differentiable on (a,b) only
• C. f continuous on [a,b] only
• D. f continuous on [a,b] and differentiable on (a,b)

Answer: (D)

The Mean Value Theorem requires that a function be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). This combination ensures a couple of key things: continuity on the whole interval guarantees the function behaves without jumps from a to b, so the secant line connecting (a, f(a)) and (b, f(b)) is meaningful, and differentiability on the interior guarantees a tangent line exists at every interior point. With those conditions, there is some c in (a,b) where the instantaneous rate of change f'(c) matches the average rate of change (f(b) − f(a)) / (b − a). If either condition is missing, the guarantee can fail: a break at an endpoint can prevent the secant slope from being matched, and a cusp or corner inside can prevent a well-defined tangent line. Therefore, the required combination is continuity on [a,b] and differentiability on (a,b).

The Mean Value Theorem requires that a function be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). This combination ensures a couple of key things: continuity on the whole interval guarantees the function behaves without jumps from a to b, so the secant line connecting (a, f(a)) and (b, f(b)) is meaningful, and differentiability on the interior guarantees a tangent line exists at every interior point. With those conditions, there is some c in (a,b) where the instantaneous rate of change f'(c) matches the average rate of change (f(b) − f(a)) / (b − a). If either condition is missing, the guarantee can fail: a break at an endpoint can prevent the secant slope from being matched, and a cusp or corner inside can prevent a well-defined tangent line. Therefore, the required combination is continuity on [a,b] and differentiability on (a,b).