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Free Rolle's theorem calculator for calculus analysis. Verify theorem conditions, find critical points, and analyze function derivatives. Our calculator checks the three essential conditions and identifies points where f'(c) = 0 according to Rolle's theorem.
Last updated: February 2, 2026
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Interval: [-2, 2] - Rolle's theorem requires a < b
Rolle's Theorem
DOES NOT APPLY
Continuous on [a,b]
✓
Differentiable on (a,b)
✓
f(a) = f(b)
✗
Analysis:
Polynomial function is continuous and differentiable everywhere
Verification:
Check endpoint equality: f(a) = f(b) for Rolle's theorem
Rolle's Theorem Conditions:
Three Conditions
Continuous, Differentiable, Equal Endpoints
Checks continuity on [a,b], differentiability on (a,b), and f(a) = f(b)
Solution Method
Solve f'(x) = 0
Identifies all points in the interval where the derivative is zero
Always Satisfies
Conditions 1 & 2
Polynomials are continuous and differentiable everywhere
Common Example
sin(x) on [0, π]
sin(0) = sin(π) = 0, critical point at x = π/2
Special Case
MVT when f(a) = f(b)
When endpoints are equal, MVT reduces to Rolle's theorem
Horizontal Tangent
Slope = 0
Guarantees at least one horizontal tangent line exists
For function f(x) = x² - 4 on interval [-2, 2]:
Theorem Status
APPLIES
Critical Point
x = 0
Our Rolle's theorem calculator systematically verifies the three essential conditions and identifies critical points where the derivative equals zero. The analysis applies fundamental calculus principles to determine theorem applicability and locate guaranteed critical points.
If f is continuous on [a, b] and differentiable on (a, b)
and f(a) = f(b)
then ∃ c ∈ (a, b) such that f'(c) = 0
Condition 1
Continuous on [a, b]
No breaks or jumps
Condition 2
Differentiable on (a, b)
Derivative exists
Condition 3
f(a) = f(b)
Equal endpoints
Shows curve with equal endpoints and horizontal tangent line
Rolle's theorem is a fundamental result in calculus that guarantees the existence of critical points under specific conditions. It serves as the foundation for the Mean Value Theorem and has important applications in optimization, root-finding, and function analysis.
Need help with other calculus concepts? Check out our derivative calculator and critical numbers calculator.
Get Custom Calculator for Your PlatformCritical Points: x = ±1/√3 ≈ ±0.577
Both points lie in the open interval (-1, 1), confirming Rolle's theorem.
f(x) = sin(x) on [0, π]
Critical point: x = π/2
f(x) = x² - 4 on [-2, 2]
Critical point: x = 0
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