TITLE: If $f$ is continuous on $[1,8]$ and some values of $f$ are given, which of the following statement concerning the existence of solutions must be true?
QUESTION [5 upvotes]: I have graphed this function, and it seems to be a parabola like figure. However, I am not sure what I am supposed to do. Am i supposed to construct the function? Or am I missing the point here? A small hint to guide me in the right direction would be appreciated! :D

REPLY [5 votes]: The function need not be a parabola.
We know that the function is continuous, try to use intermediate value theorem to make conclusion. For example, we know that $f(x)=0$ must has a solution in $[1,5]$ since $f(1)=-2$ and $f(5)=10$. and we have $-2 \le 0 \le 10$.
Also, construct counter examples of functions to show that some of them need not be true.

REPLY [3 votes]: No, you are not supposed to construct the function.(That's impossible). Since you know that $f$ is continuous my hint is to use Bolzano's Theorem.