Question

If $f\left(x\right)$ satisfies the conditions of Rolle's theorem in $[1,2]$ and $f\left(x\right)$ is continuous in $[1,2]$, then ${\int }_1^{2}f\text{'}\left(x\right)dx$ is equal to

• A. $3$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is B

$0$

Explanation for the correct option.

Find the value of ${\int }_1^{2}f\text{'}\left(x\right)dx$:

Given,

$f\left(x\right)$ satisfies the conditions of Rolle's theorem in $[1,2]$ and $f\left(x\right)$ is continuous in $[1,2]$. Then

$f\left(2\right)=f\left(1\right)$

Then by Rolle's Theorem,

$\begin{array}{rcl}{\int }_1^{2}f\text{'}\left(x\right)dx& =& f\left(2\right)-f\left(1\right)\left[\because f\left(2\right)=f\left(1\right)\right]\\ & =& 0\end{array}$

Hence, the correct option is B.