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Lagrange Mean Value Theorem Calculator

Mean-value theorem verifier. Enter f(x) on [a,b] to compute the average slope and find c with f′(c) equal to it—for calculus.

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f′(c) = (f(b)−f(a))/(b−a), c∈(a,b)
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Endpoint
Endpoint, b>a

📖 Tutorial｜Lagrange Mean Value Theorem

1. Definition
If f is continuous on [a,b] and differentiable on (a,b), some c∈(a,b) has f′(c)=(f(b)−f(a))/(b−a).
2. Symbols
SymbolMeaning
[a,b]Closed interval
f(a), f(b)Endpoint values
cMean point
f′(c)Derivative at c
3. How it works
  • We compute the secant average slope;
  • Then scan for c where f′ is closest to it;
  • This approximates the mean point.
4. Steps
  1. Enter f(x);
  2. Enter [a,b];
  3. Click Calculate.
5. Example
Example: f=x² on [0,2]. Slope=(4−0)/2=2, f′(c)=2c=2 gives c=1.
6. Pitfalls
The theorem requires continuity and differentiability;
c need not be unique;
The scanned c is approximate.

❓ FAQ｜Lagrange Mean Value Theorem

Geometric meaning?
Some tangent at c is parallel to the chord joining (a,f(a)) and (b,f(b)).
Is c unique?
Not necessarily; quadratic functions have exactly one.
What are the assumptions?
f continuous on [a,b] and differentiable on (a,b).
Relation to Rolle?
Rolle is the special case f(a)=f(b), so the slope is 0: a horizontal tangent.
Why is c approximate?
We scan a grid, so accuracy depends on grid density.