Does the function satisfy the hypotheses of the Mean Value Theorem on the given

Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = In x, [1, 6] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [1, 6] and differentiable on (1, 6). No, f is not continuous on [1, 6]. No, f is continuous on [1, 6] but not differentiable on (1, 6). 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).

Explanation / Answer

f(x) = ln(x)

Test for continuity over [1 , 6] :

We know that ln(x) is a term where the argument of the log must be greater than 0
So, domain : x > 0
Therefore, it is continuous for all positive real values and is therefore definitely continuous over [1 , 6]

Test for differentiability over (1 , 6) :

f(x) = ln(x)
So, f'(x) = 1/x ---> This is not finite only when x = 0
But our interval is (1 , 6)
So, it is differetiable over (1 , 6)

Since both the criteria are satisfied,

Yes, f is continuous on [1 , 6] and differentiable on (1 , 6) ----> OPTION TWO

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c-value :

f(b) - f(a) / (b - a) = f'(c)

f'(x) = 1/x, so f'(c) = 1/c

f(6) - f(1) / (6 - 1) = 1/c

(ln(6) - ln(1)) / 5 = 1/c

ln(6) / 5 = 1/c

So, c = 5 / ln(6) ---> This does lie within range [1 , 6]

So, the value of c = 5 / ln(6) -----> SECOND ANSWER

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