I'm new to mathematical proofs, and I have just covered the Intermediate Value Theorem. I have tried to practice my understanding of the theorem, but I have encountered a question that I'm not sure how to approach.
The question is:
Assume that $f$ and $g$ are continuous on the interval $[0,1]$ and $0 ≤ f(x) ≤ 1$ for all $x∈[0,1]$. Show that if $g(0) = 0$ and $g(1) = 1$, then there exists a $c\in[0,1]$ such that $f(c) = g(c)$.
What I have tried doing:
Introducing another function like such $h(x) = f(x) – g(x)$ and applying the theorem. However, I'm not sure if that is possible, or how I should go about doing it.
Any help would be greatly appreciated!
Best Answer
That's exactly the right thing to do!
$h$ is continuous as $f, g$ are. And $h(0) = f(0)-g(0) = f(0) - 0=f(0)$ so $0 \le h(0) = f(0) \le 1$. whereas $h(1) = f(1) -g(1) = f(1) - 1$. Now $0 \le f(1)$ so $-1 \le f(1)-1= h(1) \le 0$.
Now IVT on $h$: $h(0) \ge 0 \ge h(1)$ so there $c\in [0,1]$ so that $h(c) = 0$. Which would mean $h(c) = f(c) - g(c) = 0$ and so $f(c)=g(c)$.