Question

According to the Intermediate Value Theorem, if a polynomial has points: (0,10) and (6, -8)

Answer 1

The IVT would say that for a continuous function $f$ on an interval $[a,b]$, we can guarantee that for some value of $c\in[a,b]$ we have $f(a)\le f(c)\le f(b)$ or $f(b)\le f(c)\le f(a)$, depending on how $f$ behaves over the given interval.


If the polynomial passes through (0, 10) and (6, -8), then we know there is some $c$ such that $-8\le f(c)\le10$. In that case, only the second option makes sense. We can't claim to know anything about how $f$ behaves between values of 0 and 6, only between the values of -8 and 10, so we ignore the first and third options. The fourth scenario is possible, but not something we can guarantee because the IVT doesn't make claims about what is not possible.