Math · Calculus I · Worked example
Find the c the Mean Value Theorem promises
Find every c in (0, 2) that satisfies the Mean Value Theorem for f(x) = x³ − x on [0, 2].
Check the conditions
f is a polynomial, so it is continuous on [0, 2] and differentiable on (0, 2).
Find the secant slope
f(0) = 0 and f(2) = 6, so the average rate of change is 6/2 = 3.
Set the derivative equal to it
f′(x) = 3x² − 1, so solve 3c² − 1 = 3.
Keep the value inside the interval
c = ±2/√3, and −2/√3 is outside (0, 2). So c = 2/√3.
Result
c = 2/√3 ≈ 1.155.
Your turn
Find the c the Mean Value Theorem gives for f(x) = x² on [1, 4].
Show the answer and explanation
c = 5/2.
The secant slope is (16 − 1)/3 = 5, and f′(c) = 2c = 5 gives c = 5/2, which is in (1, 4).
Keep exploring
In Derivative Tracer, the tangent slope at x ≈ 1.155 reads 3, the secant’s slope. Near x ≈ 0.577, which is 1/√3, it reads about 0: Rolle’s theorem on [0, 1], where f(0) = f(1) = 0.
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