What is the Mean Value Theorem?
Then there is a number c in (a, b) such that
How to use the Mean Value Theorem?
Given f(x) = x3 – x, a = 0 and b = 2. Use the Mean Value Theorem to find c.
Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 2] and differentiable on (0, 2).
By the Mean Value Theorem, there is a number c in (0, 2) such that
f(2) – f(0) = f ’(c) (2 – 0)
We work out that f(2) = 6, f(0) = 0 and f ‘(x) = 3x2 – 1
We get the equation
But c must lie in (0, 2) so
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