I'm doing a review packet for Calculus and I'm not really sure what it is asking for the answer?The question is: Let f be a continuous function on the closed interval [-3, 6]. If f(-3)=-2 and f(6)=3, what does the Intermediate Value Theorem guarantee? I get that the intermediate value theorem basically means but not really sure how to explain it?

Question

I&#039;m doing a review packet for Calculus and I&#039;m not really sure what it is asking for the answer?The question is: Let f be a continuous function on the closed interval [-3, 6]. If f(-3)=-2 and f(6)=3, what does the Intermediate Value Theorem guarantee? I get that the intermediate value theorem basically means but not really sure how to explain it?

Jamiya Costa · Accepted Answer

Since   f  (  −  3  )  =  −  2  &amp;lt;  0  &amp;lt;  3  =  f  (  6  ), we can guarantee that the function has a zero in the interval [−3,6]. We cannot conclude it has only one, though (it may be many zeros).EDIT: As has already been pointed out elsewhere, the IVT guarantees the existence of at least one   x  ∈  [  −  3  ,  6  ] such that   f  (  x  )  =  c for any   c  ∈  [  −  2  ,  3  ]. Note that the fact that there is a zero may be important (for example, you couldn&#039;t define a rational function over this domain with this particular function in the denominator), or you may be more interested in the fact that it attains the value y=1 for some   x  ∈  (  −  3  ,  6  ). I hope this helps make the solution a little bit more clear.

bandikizaui · Accepted Answer

It means that for every value     c  ∈  [  f  (  −  3  )  ,  f  (  6  )  ]  =  [  −  2  ,  3  ]   there exists at least one value         x    c    ∈  [  −  3  ,  6  ]   s.t.     f  (      x    c    )  =  c  .The above, for example, tells us the function has a zero in     [  −  3  ,  6  ]  ...