`f(x) = ln(x), [1,4]` Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers `c` that satisfy the conclusion of the Mean Value...

`f(x) = ln(x), [1,4]` Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers `c` that satisfy the conclusion of the Mean Value Theorem.

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Chapter 4, 4.2 - Problem 11 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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The Mean Value Theorem requires that f be continuous on [1, 4] and differentiable on (1, 4). This is true because ln(x) is differentiable for x>0.

c mentioned is a number from (1, 4) such that f'(c)=(f(4)-f(1))/(4-1)=ln(4)/3.

f'(c)=1/c and = ln(4)/3, so `c=3/ln(4) approx 2.16.`

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