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?

Question

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?

svirajueh · Accepted Answer

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  &amp;lt;  b  &amp;lt;  6, there is an   a with   8  &amp;lt;  a  &amp;lt;  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.