Question

If $f\left(x\right)$ satisfies the condition of Rolle's theorem in $[1,2]$, then $\underset{1}{\overset{2}{\int }}{f}^{\prime }\left(x\right)dx$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. can't be determined

Answer 1

The correct option is A $0$

As $f\left(x\right)$ satisfies Rolle's theorem
$\Rightarrow f\left(x\right)$ is continuous and differentiable in $[1,2]$ and $f\left(1\right)=f\left(2\right)$
$\Rightarrow \underset{1}{\overset{2}{\int }}{f}^{\prime }\left(x\right)dx=\underset{1}{\overset{2}{\int }}\frac{d}{dx}f\left(x\right)dx=\underset{1}{\overset{2}{\int }}d\left(f\right(x\left)\right)={f\left(x\right)|}_1^2=f\left(2\right)-f\left(1\right)=0$