I have a problem with the Intermediate value theorem. For example if I have the function ( x ) = − 4 x 2 + 12 x, I can get for example all the values from x = 0 to x = 2, so f ( 0 ) = 0 and f ( 2 ) = 8, with the Intermediate value theorem, I know that the function takes all the values from 0 to 8. But it also takes the value 9 when x goes from 0 to 2, so with the Intermediate value theorem I can´t know all the value that the function takes.Can anyone explain me why this happens in this example, and obviously in other example.

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I have a problem with the Intermediate value theorem. For example if I have the function   (  x  )  =  −  4      x    2    +  12  x, I can get for example all the values from   x  =  0 to   x  =  2, so   f  (  0  )  =  0 and   f  (  2  )  =  8, with the Intermediate value theorem, I know that the function takes all the values from 0 to 8. But it also takes the value 9 when x goes from 0 to 2, so with the Intermediate value theorem I can´t know all the value that the function takes.Can anyone explain me why this happens in this example, and obviously in other example.

grcalia1 · Accepted Answer

The IVT claims that if   f  :  [  a  ,  b  ]  →      R   is continuous then   f has to take on all value between   f  (  a  ) and   f  (  b  ) for at least one   x  ∈  [  a  ,  b  ].In your example, the function achieves a extremum inside the interval, such a situation is not covered by IVT. The IVT has a limited set of assumptions, and knowing values on the boundary with guaranteed continuity is not enough to characterize all values inside that the function will take.

I have a problem with the Intermediate value theorem. For example if I have the...

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