Find all value(s) of c (if any) that satisfy the conclusion of the Mean Value Theorem for the function $f ( x ) = \frac { 1 } { 1 + x }$ on the interval $[ 0,1 ]$ .
A) $\frac { 1 } { 2 }$

B) $\frac { 1 } { \sqrt { 2 } }$
C) $\frac { 1 } { 4 }$
D) $- \frac { 1 } { 4 }$
E) $\sqrt { 2 } - 1$
F) $2 - \sqrt { 2 }$

G)0

H)No value

Question

Find all value(s) of c (if any) that satisfy the conclusion of the Mean Value Theorem for the function $f ( x ) = \frac { 1 } { 1 + x }$ on the interval $[ 0,1 ]$ .
A) $\frac { 1 } { 2 }$

B) $\frac { 1 } { \sqrt { 2 } }$ 
C) $\frac { 1 } { 4 }$ 
D) $- \frac { 1 } { 4 }$ 
E) $\sqrt { 2 } - 1$ 
F) $2 - \sqrt { 2 }$

G)0

H)No value

Quizplus · Accepted Answer

The Mean Value Theorem states that there exists a $c$ in $(0,1)$ such that $f'(c) = \frac{f(1) - f(0)}{1 - 0}$. Calculating, $f'(x) = -\frac{1}{(1+x)^2}$ and $\frac{f(1) - f(0)}{1 - 0} = -\frac{1}{2}$. Solving $-\frac{1}{(1+c)^2} = -\frac{1}{2}$ gives $c = \sqrt{2} - 1$.

Find All Value(s) of C (If Any) That Satisfy the Conclusion