CUET · MATHS · PYQ PAPER 2023

The conditions required for applicability of Rolle's theorem for a real valued function \(f(x)\) on an interval \([a, b]\) are:
(A) \(f\) is continuous on \([a, b]\)
(B) \(f\) is differentiable on \((a, b)\)
(C) \(f(a)=f(b)\)
(D) \(f^{\prime}(c)=\)
Choose the correct answer from the options given below :

A (A),(B)and(C)only
B (A),(B),(C)and(D)
C (A)and(B)only
D (A)and(D)only

Verified Solution

Answer & Solution

Correct Answer

(A) (A),(B)and(C)only

Step-by-step Solution

Detailed explanation

The conditions for Rolle's theorem are: (A) \(f\) is continuous on \([a, b]\) (B) \(f\) is differentiable on \((a, b)\) (C) \(f(a)=f(b)\) Thus, the correct answer is (A),(B)and(C)only.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match List-I with List-II
Let \(f: A \rightarrow B\) be a function given by \(f(x)=x^2\)

List - I (Domain and Co-domain)	List - II (Kind)
(A) \(A= R\) and \(B= R\)	(I) \(f\) is both one-one and onto
(B) \(A= R\) and \(B=[0, \infty]\)	(II) \(f\) is one-one but not onto
(C) \(A=B=[0, \infty]\)	(III) \(f\) is not one-one but onto
(D) \(A=[0, \infty], B= R\)	(IV) \(f\) is neither one-one nor ont

Choose the correct answer from the options given below :

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