The Intermediate Value Theorem has been proved already: a continuous function on an interval [...

flightwingsd2

flightwingsd2

Answered

2022-06-28

The Intermediate Value Theorem has been proved already: a continuous function on an interval [ a , b ] attains all values between f ( a ) and f ( b ). Now I have this problem:

Verify the Intermediate Value Theorem if f ( x ) = x + 1 in the interval is [ 8 , 35 ].

I know that the given function is continuous throughout that interval. But, mathematically, I do not know how to verify the theorem. What should be done here?

Answer & Explanation

svirajueh

svirajueh

Expert

2022-06-29Added 29 answers

I will assume that you are having trouble with the intended meaning of the question.

We have f ( 8 ) = 3 and f ( 35 ) = 6. Since f ( x ) is continuous on our interval, if follows by the Intermediate Value Theorem that for any b with 3 < b < 6, there is an a with 8 < a < 35 such that f ( a ) = b.

You are being asked to show that the Intermediate Value Theorem holds in this specific situation without using the IVT. Effectively, you are being asked to express the required a in terms of b, and to verify that it is indeed between 8 and 35.

So we want a + 1 = b. Now you can take over.

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