# `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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Expert Answers

Borys Shumyatskiy | Certified Educator

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.`