Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = ln x, [1,7] choose one letter then find c= A.Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. B.Yes, f is continuous on [1, 7] and differentiable on (1, 7). C.No, f is not continuous on [1, 7]. D.No, f is continuous on [1, 7] but not differentiable on (1, 7). E.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=

Answer 1

ANSWER

B.Yes, f is continuous on [1, 7] and differentiable on (1, 7).

$c\approx 3.08$

EXPLANATION

The given

$f(x) = ln(x)$

The hypotheses are

1. The function is continuous on [1, 7].

2. The function is differentiable on (1, 7).

3. There is a c, such that:

$f'(c) = \frac{f(7) - f(1)}{7 - 1}$

$f'(x) = \frac{1}{x}$

This implies that;

$\frac{1}{c} = \frac{ ln(7) - 0}{6}$

$\frac{1}{c} = \frac{ ln(7)}{6}$

$c = \frac{6}{ ln(7) }$

$c\approx 3.08$

Since the function is continuous on [1, 7] and differentiable on (1, 7) it satisfies the mean value theorem.