Question

If f (x) is differentiable in the interval [2, 5], where $f\left(2\right)=\frac{1}{5}$ and $f\left(5\right)=\frac{1}{2}$, then there exists a number c, 2 < c < 5 for which f ' (c) is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{5}$
• C. $\frac{1}{10}$
• D. $7$

Answer 1

The correct option is C

$\frac{1}{10}$

As f (x) is differentiable in [2 , 5], therefore, it is also continuos in [2, 5]. Hence, by mean value theorem, there exists a real number c in (2, 5) such that
$f^{\prime }\left(c\right)=\frac{f\left(5\right)-f\left(2\right)}{5-2}\Rightarrow f^{\prime }\left(c\right)=\frac{\frac{1}{2}-\frac{1}{5}}{3}=\frac{1}{10}.$
Hence (c) is the correct answer.