How to Approach a Proof Concerning Extreme Values on an Interval

Question

I've been thinking about the following problem. I would really appreciate a small push in the right direction!

Here is the problem:

"Let f be continuous on $[a,b]$, where $a < b$, such that $f(x)\ne 0$ for all $x \in [a,b]$. Prove that there is $c > 0$, such that either $f(x) > c$ for all $x \in [a,b]$ or $f(x) < -c$ for all $x \in [a,b]$."

I see two cases here, that seem to boil down to the same thing.

1) $a,b>0$. We are tasked to prove the existence of a minimum $c$.

2)$a,b<0$. We are tasked to prove the existence of a maximum $-c$, i.e. a minimum $|c|$.

From what I understand, both cases ask us to prove the existence of a minimum $|c|$.

I have learned of the applications (but not proofs) of the extreme and intermediate value theorems. I thus ask MSE how I should approach proving this result.

I suspect that I may need to apply EVT and IVT in conjunction, where I operate using terms like "$f(c)-N=0$" (taking advantage of the fact that I can assume IVT holds).

I'm really quite confused about this. I would love to hear MSE's thoughts.

Answer 1

Hint: Since $f(x)\not=0$ you may consider the function $$g(x)=\frac{1}{f(x)}$$ which is continuous on $[a,b]$. Try to use EVT on this function $g$.

Answer 2

The continuous function $x\mapsto |f(x)|$ attains its minimum on the compact interval $[a,b]$. Show that this minimum is $>0$ and let $c$ be half of it. Why does that work?