Question

How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [3, 5] for `f(x)=2sqrt(x)+3`?

Answer 1

Solve `f'(x) = (f(5)-f(3))/(5-3)` on the interval `(3,5)`.

Explanation:

`f'(x) = 1/sqrtx` and

`(f(5) -f(3))/(5-3) = sqrt5-sqrt3`.

Solve:

`1/sqrtx = sqrt5-sqrt3`.

We get

`x= (1/(sqrt5-sqrt3))^2`

`= 1/(8-2sqrt15)`

`= (4+sqrt15)/2`

Since `sqrt16` is a bit less than `4`, this `(4+sqrt15)/2` is a bit less than `(4+4)/2 = 4` and is in the interval `(3,5)`.

As an alternative at the end we could point out that Since `f` satisfies the hypotheses of MVT on `[3,5}`, it must satisfy the conclusion. Furthermore with only one candidate for `c`, this must be the `c` mentioned in the conclusion of MVT.