`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.
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.`