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Extended Mean Value Theorem
Prove the Extended Mean Value Theorem: If f and are continuous on the closed interval [a, b], and if exists in the open interval (a, b), then there exists a number c in (a, b) such that
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1. Let f (x) = x^3 + x. The Mean Value Theorem guarantees that there is a value of c on the interval 4 < x < 9 where the tangent slope (instantaneous rate of change) f ʹ(c) is the same as the secant slope (average rate of change) on that interval. By hand, work out the exact value of c where this happens.