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Does the function #f(x) = ln x# satisfy the hypotheses of the Mean Value Theorem on the given interval [1, 7]?

Answer 1

Is it continuous on the interval #[1,7]#? Yes, the #ln# is continuous on its entire domain, #(0,oo)#

Is it differentiable on the interval #(1,7)#? Yes, #f'(x) = 1/xx# exists for all #x > 0#, so it certainly exists for #x in (1,7)#.

That's it. Yes, the function satisfies the hypotheses of the Mean Value Theorem.

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Answer 2

Isaiah Allen

Yes, the function ( f(x) = \ln x ) satisfies the hypotheses of the Mean Value Theorem on the interval ([1, 7]).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.