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

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

Answer 1

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.

Answer 2

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.

Answer 3

I'm not sure if this hint is sufficiently small but consider the following problem:

Is it possible to construct a continuous function f for which f(0)=0, f(1)=1, but for which $|\{x \mid x \in [0,1], f(x)=1/2\}|=0$?

There is a theorem which can guide your thinking. The above problem should guide you toward it.

(The theorem I allude to is the IVT. Revealing this amounts to giving away the solution, but it seems that the cat is out of the bag)