CUET · MATHS · PYQ PAPER 2025

For the function \(f(x)=x^x, x>0\), which of the following are TRUE?
(A) \(f^{\prime}(x)=x^x(1+\log x)\)
(B) \(x=e\) is the critical point
(C) \(f\) is increasing in \(\left(\frac{1}{e}, \infty\right)\)
(D) \(f\) is increasing in \((0, \infty)\)
Choose the correct answer from the options given below :

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

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Answer & Solution

Correct Answer

(C) (A) and (C) only

Step-by-step Solution

Detailed explanation

\(f(x)=x^x\) \(f'(x) = x^x(1 + \log x)\) Statement (A) is TRUE. \(f'(x) = 0 \Rightarrow 1 + \log x = 0 \Rightarrow \log x = -1 \Rightarrow x = e^{-1}\) Critical point is \(x = \frac{1}{e}\). Statement (B) is FALSE.…

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.

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Match List I with List II

LIST I	LIST II
A. \(\lim_{x \to 0} \frac{(1 - \cos 2x) \sin 5x}{x^2 \sin 3x}\)	I. 18
B. \(\lim_{x \to \infty} \frac{(3x-5)(2x-7)}{(4x-9)(5x-3)}\)	II. \(\frac{10}{3}\)
C. \(\lim_{x \to 0} \frac{2 \sin^2 3x}{x^2}\)	III. \(\frac{3}{4}\)
D. \(\lim_{x \to \frac{\pi}{4}} \frac{1 - \cos^3 x}{2 \cot x - \cot^3 x}\)	IV. \(\frac{3}{10}\)

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CUET 2023 Hard

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For the function \(f(x)=x^x, x>0\), which of the following are TRUE?(A) \(f^{\prime}(x)=x^x(1+\log x)\)(B) \(x=e\) is the critical point(C) \(f\) is increasing in \(\left(\frac{1}{e}, \infty\right)\)(D) \(f\) is increasing in \((0, \infty)\)Choose the correct answer from the options given below :

Step-by-step Solution