Question

How do you find all numbers c in the interval show that the function `f(x)=x^3+x+1` satisfies the mean value theorem on the interval [-1,2]?

Answer 1

The secant of `f(x)=x^3+x+1` over the interval `[-1,2]`
is
`(f(2) - f(-1))/(2- (-1))`
`= (11- (-1))/(2+1) = 4`

According to the Mean Value Theorem there should exist at least one value `c` such that
`f'(c) = 4` for this interval.

Since `f'(x) = 3x^2+1`
we are looking for one (or more values) that satisfy
`3c^2+1=4`

`rarr c = +-1`

Within the interval `[-1,2]` the Mean Value Theorem is satisfied by
`c=-1` and `c=1`