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?

1 Answer
Jun 17, 2015

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 cs 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