# 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?

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 $c \in \left(a , b\right) : f \left(b\right) - f \left(a\right) = f ' \left(c\right) \left(b - a\right)$

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

So we find them out:

$f \left(0\right) - f \left(1\right) = f ' \left(c\right) \left(0 - 1\right) \implies f ' \left(c\right) = \frac{1}{7}$

i.e.

$\frac{1}{7} = \frac{\left(c + 6\right) - c}{c + 6} ^ 2 \implies {\left(c + 6\right)}^{2} = 42 \implies {c}_{1} = - 6 + \sqrt{42} , {c}_{2} = - 6 - \sqrt{42}$

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 ${c}_{1} \in \left(0 , 1\right)$ without any kind of manual verification