Let f(x) = |x| on the interval [-1, 1]. A student claims the MVT applies because f is continuous and the average rate of change from -1 to 1 is (f(1)-f(-1))/(1-(-1)) = 0, so there must exist c where f'(c) = 0. What is wrong?
ANothing — f is continuous on [-1,1], so MVT applies and the conclusion is correct
BThe error is in computing the average rate of change; f(-1) ≠ f(1)
Cf is not differentiable at x = 0, violating the differentiability hypothesis on the open interval (-1, 1), so MVT does not apply
DMVT only applies to strictly increasing or decreasing functions
The MVT requires both continuity on the closed interval [a,b] AND differentiability on the open interval (a,b). While f(x) = |x| is continuous everywhere, it fails to be differentiable at x = 0, which lies in the open interval (-1, 1). A single point of non-differentiability is enough to invalidate the theorem. And indeed, the conclusion fails: |x| has no point with f'(c) = 0 — the derivative is +1 for x > 0 and -1 for x < 0. Checking hypotheses is not a formality.
Question 2 Multiple Choice
Which of the following is the most important use of the MVT in theoretical calculus?
AFinding the exact value of c where the instantaneous rate equals the average rate
BComputing definite integrals by finding average values of functions
CProving that if f'(x) = 0 for all x on an interval, then f is constant on that interval
DDetermining the slope of the tangent line at a given point without using limits
The MVT's main power is as a proof tool, not a computational one. The corollary that zero derivative implies constant function is the theorem's most load-bearing application: it underpins the uniqueness part of antiderivative theory, justifies the First Derivative Test for increasing/decreasing behavior, and is a key step in proving L'Hôpital's Rule. Finding the actual value of c (option A) is rarely the point — the theorem usually just guarantees c exists, which is sufficient for the proofs that follow.
Question 3 True / False
The MVT guarantees that there is exactly one point c in (a, b) where the instantaneous rate of change equals the average rate of change.
TTrue
FFalse
Answer: False
False — the MVT guarantees at least one such point c, not exactly one. A function can have many points where the tangent is parallel to the secant, especially if it oscillates. For example, f(x) = sin(x) on [0, 4π] has the same endpoint values (both 0), and horizontal tangents occur at x = π/2, 3π/2, 5π/2, and 7π/2 — four points, not one. The theorem is an existence result, not a uniqueness result.
Question 4 True / False
The Mean Value Theorem and the Intermediate Value Theorem are essentially the same result applied to different contexts.
TTrue
FFalse
Answer: False
False — they address completely different properties of functions. The IVT says: if f is continuous on [a,b] and k is between f(a) and f(b), then f takes the value k somewhere in (a,b). It concerns function values. The MVT says: if f is continuous on [a,b] and differentiable on (a,b), then f'(c) equals the average rate of change for some c. It concerns derivatives. They share the continuity hypothesis but prove different things about different aspects of the function.
Question 5 Short Answer
A car travels 120 miles in exactly 2 hours. Using the MVT, what can you conclude, and what hypotheses are needed for the conclusion to hold?
Think about your answer, then reveal below.
Model answer: The MVT guarantees that at some moment during the 2-hour trip, the car's instantaneous speed was exactly 60 mph — the average rate of change of position. The hypotheses needed are: (1) the position function f(t) is continuous on [0, 2] (the car doesn't teleport) and (2) f is differentiable on (0, 2) (the car has a well-defined instantaneous velocity at every interior moment). The theorem does not say when during the trip the speedometer read 60, nor that it happened only once — only that it must have happened at least once.
This is the 'speedometer theorem' intuition that makes MVT memorable. The average speed over any interval must be achieved as an instantaneous speed at some point. The hypotheses are physical reasonableness conditions — continuity rules out teleportation, differentiability rules out infinite acceleration. Both are realistic for a car.