Question

Question #be0d7

Answer 1

Please see the explanation section, below.

Explanation:

The Intermediate Value Theorem (IVT) is a theorem about functions that are continuous on closed interval.

(One version of) IVT says that

If `f` is continuous on `[a,b]` with `f(a) != f(b)` and `M` is between `f(a)` and `f(b)` (exclusive),

then there is a `c` in `(a,b)` with `f(c)=M`

In order to use the IVT, we must have a function that is continuous on a closed interval as the topic of discussion.

One use for the IVT is to show that a function takes on a particular value.

Any solution to `3 = x^2 + sin(pix)` is also a solution to `x^2 + sin(pix)-3=0`, and vice versa.

To use IVT to show that the equation `3 = x^2 + sin(pix)` has a solution, we can

(1) use IVT to show that the function: `f(x) = x^2+sin(pix)` takes on the value `3`

OR

(b) use IVT to show that the function: `f(x) = x^2+sin(pix)-3` takes on the value `0`

I have a slight preference for method (b). So we'll show that `f(x) = x^2+sin(pix)-3` has a (real number) zero.

We now need an interval `[a,b}` on which `f` is continuous. And, since out "target" number `M` is `0`, we want one of `f(a)` and `f(b)` to be positive and the other negative.

For integer `x`, `sin(pix) = 0`, so `f(x) = x^2-3`

`f(1) = -2` is negative and `f(2)= 1` is positive.

Now we're ready to write the proof.

Proof

`x` is a solution to `3 = x^2 + sin(pix)` if and only if `x` is a zero of `f(x) = x^2+sin(pix)-3`.

`f` is continuous on `[1,2]` because it is the sum of continuous functions.
The function, `x^2-3` is a polynomial, so it's continuous everywhere. And `sin(pix)` is the composition of the sine function with `pix` (a linear function), so it is also continuous everywhere.

`f(1) = -1` and `f(2) = 1`, so `0` is (strictly) between `f(1)` and `f(2)`.

Therefore, by the Intermediate Value Theorem, there is a `c` in `(1,21)`, such that `f(c) = 0`.

This `c` is a solution for the equation `3 = x^2 + sin(pix)`.

Important Note

Students sometimes think that the Intermediate Value Theorem tells us how to solve an equation. It does not. It sometimes tells us that a solution must exist, but it does not tell us how to find a solution.