Question

How do you verify that the function `f(x)=x/(x+6)` satisfies the hypotheses of The Mean Value Theorem on the given interval [0,1] and then find the number c that satisfy the conclusion of The Mean Value Theorem?

Answer 1

The MVT states that if f(x) is continuous in [a,b] (it obviously is) and derivable in (a,b) (it obviously is too), then `exists` at least one `c in (a,b) : f(b)-f(a)=f'(c)(b-a)`

Notice the theorem doesn't give you the number of `c`s nor their values.

So we find them out:

`f(0)-f(1)=f'(c)(0-1) => f'(c)=1/7`

i.e.

`1/7=((c+6) - c)/(c+6)^2 => (c+6)^2=42 => c_1=-6+sqrt(42), c_2=-6-sqrt(42)`

We notice `c_2<0`, so `c_2` is not a root to be considered for MVT, the only choice we have left is `c_1`, and MVT assures us `c_1 in (0,1)` without any kind of manual verification