Question

How do you use the Intermediate Value Theorem to show that the polynomial function `x^3-3x+4.1=0` has a root on the interval (0,1)?

Answer 1

Function `y=x^3-3x+4.1` has no roots in the interval `(0,1)`.

Explanation:

Lets find 1st derivative of the function:
`y'=3x^2-3`
`AAx in (0,1): 0< x^2<1` and hence `3x^2-3<0`

So, `AAx in (0,1): y'<0` and function is strictly decreasing. Maximum and minimum values of the function are, respectively:
`f_max=f(0)=4.1`
`f_min=f(1)=2.1`
`AAc in (0,1): f(c) in (2.1,4.1)`