Question

Given below are two statements:
Assertion (A): If a function \(f(x)\) is continuous in \([a,b]\) and differentiable in \((a,b)\), then there exists a point \(c\in(a,b)\) such that \(\dfrac{f(b)-f(a){b-a}=f'(c)\).
Reason (R): This is the statement of Cauchy's mean value theorem.}

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

\(\frac{f(b)-f(a)}{b-a}=f'(c)\) is Lagrange's Mean Value Theorem, not Cauchy's Mean Value Theorem.

Answer 1

The Correct Option is C

Concept:
The assertion is the statement of Lagrange's Mean Value Theorem.

Step 1: Check Assertion.
Lagrange's Mean Value Theorem states that if \(f(x)\) is continuous in \([a,b]\) and differentiable in \((a,b)\), then: \[ \frac{f(b)-f(a)}{b-a}=f'(c) \] for some: \[ c\in(a,b) \] So Assertion is correct. \[ A \text{ is correct} \]

Step 2: Check Reason.
The reason says this is Cauchy's mean value theorem. But Cauchy's mean value theorem involves two functions \(f(x)\) and \(g(x)\), and has the form: \[ \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)} \] So Reason is not correct. \[ R \text{ is incorrect} \] \[ \therefore \text{Correct Answer is (C)} \]