Question

Given the function `f(x)=x/(x+9)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,18] and find the c?

Answer 1

The Mean Value Theorem has two hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`

H2 : `f` is differentiable on the open interval `(a,b)`.

In this question, `f(x) = x/(x+9)` , `a=1` and `b=18`.

H1 is true because this function is a rational function and rational functions are continuous on their domains. So `f` is continuouos except at `x=-9` which is not in `[1,18]`

H2 is true because `f'(x) = 9/(x+9)^2` exists for all `x` except `x=-9` which is not in `(1,18)`

The conclusion of MVT says there is a number `c` in `(a,b)` such that `f'(c) = (f(b)-f(a))/(b-a)`. To find the `c`, solve the equation. Discard any solutions outside the interval `(a,b)`.

So we need to solve

`9/(x+9)^2 = ((18/27)-(1/10))/17` on `(1,18)`.

I get `-9+3sqrt30`