Question
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = ln x, [1, 9] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [1, 9] and differentiable on (1, 9). No, f is not continuous on [1, 9]. No, f is continuous on [1, 9] but not differentiable on (1, 9). There is not enough information to verify if this function satisfies the Mean Value Theorem. If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma-separated list. If it does not satisfy the hypotheses, enter DNE). c =
Explanation / Answer
f(x) is continuous in [1,0]
f'(x) = 1/x
differentiable in [1,9]
f(a) = ln1 = 0
f(b) = ln9
f(a) != f(b)
rolle's theorem cannot be used
use Lagrange's mean value theorem
f'(c) = f(b) - f(a) /(b-a)
1/c= ln9 - 0 / (9-1)
1/c= ln(9) /8
c = 8/ln(9)