Question

How do you use the Intermediate Value Theorem to show that the polynomial function `f(x) = x^4 -10x^2 + 3` has one zero?

Answer 1

Explanation below :)

Explanation:

The intermediate value theorem states that if `f` is a continuous function, and there exist two points `x_0` and `x_1` such that `f(x_0)=a` and `f(x_1)=b`, then `f` assumes every possible value between `a` and `b` in the interval `[x_0,x_1]`.

For example, if `f(3)=8` and `f(7)=10`, then every possible value between `8` and `10` is reached for `3\le x \le 7`.

This theorem is often used to find zeroes of functions: if the function you're working with is continue, and you can find two points `x_0` and `x_1` such that `f(x_0)<0` and `f(x_1)>0`, then the theorem states that every possible intermediate value is taken, zero included. Note that this theorem doesn't give you any information about the root, it only tells you that there is (at least) one root.

So, in your case, we need to find two points with image, respectively, negative and positive. Unfortunately, there is no algorithm to do so, you have to train your instinct a little bit. In this case for example, `f(0)=3`, so it's positive, while `f(1)=1-10+3=-6`, which is negative.

We're done, since `f` is a polynomial, it is continuous, and then the theorem ensures that there is a zero between `0` and `1`.

I'll post the graph of the function only for checking purpose.

graph{x^4-10x^2+3 [-2.738, 2.737, -1.366, 1.372]}