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) = ln x,    [1, 7]

___Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem.

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

____No, f is not continuous on [1, 7].

___No, f is continuous on [1, 7] but not differentiable on (1, 7).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

Since the function f(x) = lnx is continous on [1,infinity) and so it is continous on [1,7] which is subset of [1,infinity)

Also the derivative of lnx = 1/x and for [1,7] it is defined, hence f(x) = lnx is differentiable on (1,7)

Second option is correct:)

Now we can find a c such that f'(c) = (f(b)-f(a))/(b-a)

Here a=1 and b=7 and f'(c) = 1/c

So we get 1/c = (ln(7)-ln(1))/(7-1) = (ln(7)-0)/6 =( 1/6)ln(7)

And so c= 6/ln(7) or c= 3.0834

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