How do I find the numbers c that satisfy the Mean Value Theorem for f(x)=x/(x+2) on the interval [1,4] ?

1 Answer
Wataru
Sep 12, 2014

The mean value theorem guarantees that there exists a number c in (1,4) such that
f'(c)={f(4)-f(1)}/{4-1}.
The actual value of c is -2+3sqrt{2}.

Let us find the left-hand side of the above equation,
By Quotient Rule,
f'(x)={1cdot(x+2)-xcdot1}/{(x+2)^2}=2/(x+2)^2
Rightarrow f'(c)=2/(c+2)^2

Let us find the right-hand side,
{f(4)-f(1)}/{4-1}={4/6-1/3}/{3}=1/9

By setting the left-hand side and the right-hand side equal to each other,
2/(c+2)^2=1/9

by taking the reciprocal,
(c+2)^2/2=9

by multiplying by 2,
(c+2)^2=18

by taking the square-root,
c+2=pm sqrt{18}=pm3sqrt{2}

by subtracting 2,
c=-2 pm3sqrt{2}

Since 1 < c < 4,
c=-2+3sqrt{2}

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