Intermediate Value Theorem without an interval?

Question

Final Soon. I'm pretty sure the Professor said there would be an intermediate value theorem question asking to prove a function has a zero without a given interval.

I tried thinking of how this can be done but we never did that and I cant think of a way to do it. I thought maybe critical points and test points around it but I dont think that would work.

Answer 1

Here's an example of how that might go:

• Problem: Show that the function $f(x)=x^{17}-48x^5+x^2-3$ has a zero. Note that I have no clue how to actually find a zero for this function.

• Solution: Plugging in $x=1$ gives a negative value (namely, $-49$) while plugging in $x=-1$ gives a positive value (namely, $45$). By the intermediate value theorem, somewhere on the interval $[-1,1]$ we have $f(x)=0$.

Note that we've found the interval ourselves. So part of the problem, in fact, is producing that bit of information.

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We can even solve problems of this type without finding any specific interval at all. One basic, and quite useful, theorem about polynomials is the following:

Suppose $p$ is an odd-degree polynomial with positive leading coefficient (e.g., $17x^5-12435235x^2+3$). Then $\lim_{x\rightarrow-\infty}p(x)=-\infty$ and $\lim_{x\rightarrow\infty}p(x)=\infty$.

This immediately tells us that any odd degree polynomial with positive leading coefficient has a zero: by the theorem, we can find $a$ and $b$ with $p(a)<0$ and $p(b)>0$ (pick $a$ "really small" and $b$ "really big"), and so by the intermediate value theorem $p$ has a zero between $a$ and $b$; and this argument works even though we have no idea what good examples of $a$ and $b$ might be!

(And note that the positivity of the leading coefficient matters for the values of the limits at infinity, but not for the existence of a zero: if the leading coefficient of $p$ is negative, think about $-p$.)

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Basically, you're being asked to be a bit creative with the intermediate value theorem: given a function, think about what you know about the function (or can find out about it) and see if you can notice some facts which will let you set up a situation where you can use the intermediate value theorem. In this case there's not much variety: you'll always be looking for one point where it's negative and one point where it's positive, or at least for an argument that such points need to exist even if you can't find them exactly. But in more general situations, applying a given theorem isn't always easy. Indeed, it's often not even clear that a given theorem is even relevant!