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
Alberto M.
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 #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

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