Question

y 20 EXAMPLE 3 To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x3 – X, a = = 0, b = 2. 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). Therefore, by the Mean Value Theorem, there is a number c in (0, 2) such that f(2) – f(0) = f'(C)(2 - 0). 15 10 Now f(2) = f(0) = and f'(x) = so this equation becomes 5 I f'(c)(2) = )2 = II Video Example which gives c2 = that is, c = But c must be in (0, 2), so C = The figure illustrates this calculation: The tangent line at this value of c is parallel to the secant line. Need Help? Read It Talk to a Tutor

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