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Can the mean value theorem be applied to #f(x) = 2(sqrt x) + x# on the interval [1,4]?

Answer 1

Emma Aldridge

Yes it can.

Hypothesis 1: Is #f# continuous on the closed interval #[1, 4]#?

Yes, it is continuous on its domain (#x>=0#.)

Hypothesis 2 Is #f# differentiable on the open interval #(1, 4)#?

Yes. #f'(x) = 1/sqrtx +1# exists for all #x>0#, so it exists in #(1, 4)#.

That's all that is required to apply the Mean Value Theorem.

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Answer 2

Paisley Johnson

Yes, the mean value theorem can be applied to ( f(x) = 2\sqrt{x} + x ) on the interval ([1,4]). The function ( f(x) = 2\sqrt{x} + x ) is continuous on the closed interval ([1,4]) and differentiable on the open interval ((1,4)), as it is composed of elementary functions which are continuous and differentiable over their respective domains. Therefore, by the mean value theorem, there exists at least one value ( c ) in the open interval ((1,4)) such that the derivative of ( f(x) ) evaluated at ( c ) is equal to the average rate of change of ( f(x) ) over the interval ([1,4]).

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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.