Applying MVT to √x on [1,4]

Real Analysis · Axiom Academy

EXAMPLE Applying the MVT to on [1,4] Find the value c guaranteed by the Mean Value Theorem for on the interval [1,4] . Let on [1,4] . The Mean Value Theorem guarantees a point c in (1,4) where the instantaneous rate f'(c) equals the average rate of change across the interval. Verify the hypotheses, then find that c . The secant joining (1,1) and (4,2) has slope . The MVT guarantees a point c in (1,4) where the tangent to is parallel to that secant — here . Nice work! You applied the Mean Value Theorem end to end: verify the hypotheses, match the average rate, then solve for the guaranteed point. Verify the hypotheses first: is continuous on [1,4] and differentiable on (1,4) , so the MVT applies. The MVT matches two slopes: the average rate is the secant slope; the MVT guarantees a tangent with that same slope. Solve f'(c) = average rate: gives . Confirm the answer: lies in (1,4) , so c is valid and the tangent there is parallel to the secant. The Mean Value Theorem connects average and instantaneous rates of change, and it underlies many results in analysis — including the proof of the Fundamental Theorem of Calculus.

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