Math · Calculus I · Worked example
Apply Rolle’s theorem
Check that Rolle’s theorem applies to f(x) = x³ − 4x on [−2, 2], and find every c it promises.
Check the conditions
f is a polynomial, so it is continuous and differentiable everywhere, and its endpoint values are equal.
Solve f′(c) = 0
f′(x) = 3x² − 4.
Both values count
c = ±2/√3 ≈ ±1.155, and both lie in (−2, 2): the graph has a local maximum and a local minimum there. The theorem promises at least one c; here there are two.
Result
c = −2/√3 and c = 2/√3, about ±1.155.
Your turn
Find the c from Rolle’s theorem for f(x) = x² − 6x + 5 on [1, 5].
Show the answer and explanation
c = 3.
f(1) = 0 = f(5), and f′(c) = 2c − 6 = 0 gives c = 3.
Keep exploring
In Graph, the two horizontal tangents sit at the turning points, near (−1.155, 3.079) and (1.155, −3.079).
Return to the concept →Sources and scope
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Open the example inputs, change a value and keep a useful result on your board.
See the horizontal tangents in Graph Check the endpoints in Math Open worked example on a board Mean Value Theorem in Math ReferenceYour existing work stays on this device. Examples open as editable copies.