Question

If \(f(x)\) satisfy the condition of Rolle's Theorm in \([1,2]\) and \(f(x)\) is continuous in \([1,2]\) then \(\int _{1}^{2}f'(x)dx\) is equal to _____

• A. \(3\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Rolle's theorem gives equal function values at the end points, and the integral of f dash is the difference of f values.

Answer 1

The Correct Option is B

Step 1: Conditions
Rolle's theorem on $[1,2]$ means $f(1)=f(2)$.

Step 2: Integrate
An antiderivative of $f'(x)$ is $f(x)$, so the integral is $[f(x)]_{1}^{2}$.

Step 3: Evaluate
$f(2)-f(1)=0$, which matches the second option.

Final Answer:
The value of the integral is 0. \[ \boxed{\text{(B) }0} \]