Let be a differentiable function such that: with and . Consider the following two statements: (A) : for all (B) : for all

Both the statements (A) and (B) are true. Explanation: Step 1: Inequality ko simplify karna Dhyan se dekhne par, left side ek derivative ban raha hai: Ye bilkul wahi term hai jo inequality mein di gayi hai. Isliye: Step 2: Integration Dono taraf se tak integrate karte hain: Hame pata hai , toh: Step 3: Statements check karna Statement (B) ke liye: par check karte hain: Diya gaya hai . Chunki mein function ki value calculate karne par ye hamesha se badi aati hai. Isliye Statement (B) sahi hai . Statement (A) ke liye: check karne ke liye hum maximum value dekhte hain. Diye gaye conditions ke mutabiq is range mein se chhota hi rehta hai. Isliye Statement (A) bhi sahi hai .

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $f : [2, 4] \to \mathbb{R}$ be a differentiable function such that:

(x \log_e x) f'(x) + (\log_e x) f(x) + f(x) \ge 1, x \in [2, 4]

Consider the following two statements:

(A) : $f(x) \le 1$ for all $x \in [2, 4]$

(B) : $f(x) \ge \frac{1}{8}$ for all $x \in [2, 4]$

JEE 2023 — Mathematics PYQ

Choose the correct answer:

Explanation

Step 1: Inequality ko simplify karna

Step 2: Integration

Step 3: Statements check karna

Explanation

Step 1: Inequality ko simplify karna

Step 2: Integration

Step 3: Statements check karna

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