Mean Value Theorem Calculator
Verify Rolle's theorem, Lagrange's mean value theorem or Cauchy's mean value theorem on an interval: check the conditions, then find every c where the tangent is parallel to the chord.
About the Mean Value Theorem Calculator
Lagrange's mean value theorem: if f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f′(c) = [f(b) − f(a)] ÷ (b − a), so the tangent there is parallel to the chord. Rolle's theorem is the case f(a) = f(b), where f′(c) = 0.
Cauchy's form compares two functions: f′(c)/g′(c) = [f(b) − f(a)]/[g(b) − g(a)].
The formula
f′(c) = [f(b) − f(a)] ÷ (b − a), a < c < b
Worked example
f(x) = x³ − 3x + 1 on [0, 2]
- Chord slope = (3 − 1)/2 = 1
- 3c² − 3 = 1, c = 2/√3 = 2√3/3 ≈ 1.1547
Things to keep in mind
- Continuity and differentiability are tested on a grid of points, which can miss a single bad point; |x| at 0 and 1/x are caught.
Questions people ask
Can there be more than one c?
Yes. The theorem promises at least one; all of them are listed.
When does the mean value theorem apply?
f must be continuous on [a, b] and differentiable on (a, b). Then some c between a and b has f′(c) equal to the average slope (f(b) − f(a)) ÷ (b − a).