4.1.27 · D5Calculus I — Limits & Derivatives
Question bank — Mean Value Theorem — proof, Rolle's theorem
The two roles to keep straight:
- Hypotheses = the "entry ticket" conditions H1 and H2 (plus H3 for Rolle). Break one, and the theorem promises nothing.
- Conclusion = the guarantee: there exists at least one interior point where the tangent slope equals the secant slope (or for Rolle).
The traps split by which of these you mishandle.
True or false — justify
on has , so Rolle guarantees some with .
False. is not differentiable at , so hypothesis 2 (H2) fails; indeed never hits . The equal-endpoints condition alone is not enough.
If for some interior , then must have satisfied the MVT hypotheses.
False. The conclusion can hold "by luck" even for badly-behaved functions. The theorem says hypotheses conclusion, not the reverse — a true conclusion never certifies the hypotheses.
The MVT guarantees exactly one point .
False. It guarantees at least one. A wavy curve can have many tangents parallel to the secant; e.g. on a long interval.
If then the secant slope is , so Rolle is just MVT with a horizontal secant.
True. Setting makes , so the MVT conclusion becomes Rolle's conclusion. Rolle is the flat special case.
A function differentiable on all of automatically satisfies the MVT hypotheses on any .
True. Differentiable everywhere implies continuous everywhere, so both H1 (continuity on ) and H2 (differentiability on ) hold on every .
If for all in , then is constant on .
True (needs MVT). For any two points, MVT gives , so all values are equal. This is how we prove "zero derivative ⇒ constant" — see Increasing and Decreasing Functions.
The point in Rolle's theorem can equal or .
False. The conclusion places strictly inside the open interval . Differentiability is only guaranteed inside, and the flat-tangent point is claimed inside.
If is continuous on but differentiable only on except at one interior point, MVT still applies.
False. MVT requires differentiability at every interior point. One bad interior point breaks hypothesis 2 (H2) — exactly why fails.
Spot the error
" on has , so by Rolle some has ."
Error: is discontinuous at (it blows up), so it is not continuous on . Hypothesis 1 (H1) fails and no flat tangent exists.
" on satisfies , so Rolle gives ."
Error: is undefined at (vertical tangent, a cusp), so is not differentiable on the whole open interval (H2 fails). The theorem doesn't apply.
"By MVT on over , there is a with ."
Error: is not even defined (let alone continuous) at , so H1 fails. The interval must lie in the domain where hypotheses hold; you cannot straddle a discontinuity.
"MVT lets me write and solve for the exact formula in general."
Error: MVT is an existence statement — it names no formula for . For specific functions you can solve for , but there is no universal expression, and there may be several valid .
" (floor) is constant between integers, so Rolle applies on with ."
Error: Floor jumps at inside , so it is discontinuous there — hypothesis 1 (H1) fails. Being flat between jumps doesn't rescue continuity across them.
"Since MVT gives , if the average rate is positive then everywhere on ."
Error: MVT only supplies one point with slope equal to the average. Elsewhere the derivative can be negative; the function need only rise on average, not monotonically.
Why questions
Why does the proof of Rolle start by invoking the Extreme Value Theorem?
We need a guaranteed highest or lowest point to place a flat tangent. EVT says a continuous function on a closed bounded interval actually attains a max and min — that "top/bottom" is what we differentiate at.
Why must at least one extremum be interior in Case B of Rolle's proof?
Because : if both max and min occurred only at the endpoints, then , forcing a constant — contradicting . So one extreme is strictly inside.
Why does an interior maximum force rather than merely ?
Fermat's Theorem (interior extrema): approaching from the right gives , from the left . Differentiability makes both one-sided limits equal, squeezing to exactly .
Why does the MVT proof subtract the secant line to build the helper ?
Write the secant line as — the straight line through the two endpoints. Then is the vertical gap to that line, and , so H3 holds and Rolle applies. It "tilts the graph flat."
Why is still differentiable when is?
Here is a straight line, which is differentiable everywhere; a differentiable function minus a differentiable line is differentiable. So inherits 's smoothness exactly.
Why does the MVT immediately prove is Lipschitz?
MVT turns the difference into ; since , we get . A bounded derivative always yields a Lipschitz constant.
Why can't we use MVT to compare at two points if has a jump between them?
A jump is a discontinuity, breaking hypothesis 1 (H1) on that interval. Without continuity the "average slope" need not be achieved — think of a step that instantly climbs with no in-between slope.
Why is Cauchy's Mean Value Theorem "more general" than the ordinary MVT?
It compares two functions' changes, ; setting recovers the ordinary MVT. It's the engine behind L'Hôpital's rule and Taylor's Theorem remainders.
Edge cases
What does Rolle say if is constant on ?
Every hypothesis holds and everywhere, so any interior works. The theorem still delivers a — just not a unique one.
What happens to MVT when ?
The interval degenerates to a point and the secant slope is undefined. The theorem requires a genuine interval , so this case is excluded by hypothesis.
If is continuous on and differentiable on but has a vertical tangent (infinite slope) somewhere inside, is it differentiable there?
No. A vertical tangent means the derivative limit is , not a finite number, so is not differentiable at that point and H2 fails there.
Can MVT hold on an unbounded interval like ?
The classical statement needs a closed bounded interval so EVT (used in Rolle) applies. You apply MVT on each finite ; the endpoints must be finite.
If a smooth function's secant slope is but it is strictly increasing then decreasing, where is ?
At the turning point (the peak), where . Equal endpoints force at least one interior max or min, and that is the flat tangent Rolle promises.
For on , the secant slope is ; does the MVT exist and where?
Yes. gives , both inside . Here two valid points exist — a reminder that need not be unique.