Intermediate Value Theorem

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.*x*^{4} + *x* − 9 = 0, (1, 2)

*f*(*x*) = *x*^{4} + *x* − 9

is ___________on the closed interval [1, 2], *f*(1) = _________, and *f*(2) = _____________.

Since −7 < _________ < 9, there is a number *c in **(1, 2)** such that **f(c) = *____________

* by the Intermediate Value Theorem. Thus, there is a *______________* of the equation **x*^{4} + x − 9 = *in the interval (**1**, **2**).*

What is the intermediate value theorem?

Re: Intermediate Value Theorem

IVT: If $\displaystyle f: [a, b] \rightarrow \mathbb{R}$ is continuous, and t is between f(a) and f(b), then there exists c in (a, b) such that f(c) = t.

What it means is that a continuous function can't "miss" any values.

Ex: Imagine this conversation:

Farmer: "I'm angry at you, Mr Weather Service spokesman. The temperature dropped from 60 degrees to 29 degrees in one day. I had an app set up with the Weather Service to text me if the temperature ever hit 45 degrees, so that I could prepare to defend my crops from the unusal cold. But I never got any text message."

Weather Service spokesman: "Don't blame us. The temperature never hit 45 degrees."

Farmer: "Nonsense. The temperature dropped from 60 to 29, so it MUST have been 45 SOMETIME during that drop. That's because temperature changes CONTINUOUSLY. It didn't skip or leap over any values during its drop from 60 to 29!"

Re: Intermediate Value Theorem

Quote:

Originally Posted by

**Oldspice1212** What is the intermediate value theorem?[/COLOR]

Due to differences in presentation between different sources, it would be better if you consult your textbook or lecture notes. If you don't have them, see IVT on Wikipedia.