Question

How do you determine all values of c that satisfy the mean value theorem on the interval [1,9] for `f(x)=x^-4`?

Answer 1

`c=root(4)(17496/728) ~=2.2141`

Explanation:

Based on the mean value theorem, as `f(x) = x^(-4)` is continuous in `[1,9]` then there must be at least one point `c in [1,9]` for which:

`c^(-4) = 1/8int_1^9 x^(-4)dx`

As f(x) is strictly decreasing in the interval [1,9], there can be only one point in the interval satisfying the theorem.

Evaluate the integral:

`1/8int_1^9 x^(-4)dx = 1/8[-1/(3x^3)]_1^9 =1/8(1/3 - 1/2187)=1/8((729-1)/2187)=728/17496`

So the value `c` that satisfies the mean value theorem is:

`c^(-4) = 728/17496`

`c=root(4)(17496/728) ~=2.2141`