Question

How do you use the Intermediate Value Theorem to show that the polynomial function `F(x)=x^3+2x+1` has a root in the interval [-2, 1]?

Answer 1

You put the borders of the interval in the function.

Explanation:

Let's try to put the borders of the interval in the function:
`F(-2) = -8 - 4 + 1 = -11 < 0`
`F(1) = 1+2+1=4 > 0`

Since you have one function value between that is negative and one that is positive and since the function is continuous in `[-2,1]` (a polynomial is continuous everywhere), we can conclude that the function must have gone trough `y=0`. To go from a negative to a positive you have to go trough this.

You can easily see this if you plot the function:
graph{y=x^3+2x+1 [-3,2,-12,5]}

You can see that the function needs to go trough `y=0` to get from `x=-2` to `x=1`