Exams › JEE Advanced › Maths
Consider the function f(x) defined as f(x) = lim n→∞ {n(x + n)(x + n/2)...(x + n/n) / n!(x² + n²)(x² + n²/4)...(x² + n²/n²)}^x/n, where x > 0. Which of the following is true?
- f(1) is less than or equal to 1/2
- f(2/3) is greater than or equal to 1/3
- The derivative of f at x = 2 is non-positive
- The derivative of f at x = 3 is greater than or equal to the derivative of f at x = 2
Correct answer: f(2/3) is greater than or equal to 1/3
Solution
The function f(2/3) is greater than or equal to 1/3, which can be determined by analyzing the given function f(x) and its behavior at x = 2/3.
Related JEE Advanced Maths questions
- Evaluate f(x) = lim x→∞ (x²n − 1) / (x²n + 1). Which of the following is true?
- Determine the value of L if L = lim x→0 [(sin x + ae^x + be^−x + c ln(1 + x)) / x³], given that L is finite and not infinite.
- For a positive integer n, let f(n) be defined as n plus the sum of terms in the form (a + bn − cn²) / (dn + en²), where the coefficients vary across terms. Specifically, f(n) = n + (16 + 5n − 3n²) / (4n + 3n²) + (32 + n − 3n²) / (8n + 3n²) + (48 − 3n − 3n²) / (12n + 3n²) +... + (25n − 7n²) / (7n²). Determine the value of lim n→∞ f(n).
- Consider the function f: (0,1) → R defined by f(x) = √n whenever x lies in the interval [1/(n+1), 1/n), where n belongs to the set of natural numbers. Let g: (0,1) → R be a function satisfying the inequality ∫x¹ (1−t)/t dt ≤ g(x) < 2√x for all x in (0,1). Determine the value of lim x→0 f(x)g(x).
- Let h(x) = h(2x) for every real x, where h is a continuous function and h(2012) = pi/2. Define the limit M = lim_(x->0) [cos²(h(x)) + 1 - sin³(h(x))] / sin²(x). What is the value of 4M?
- Let alpha = lim(n->inf) (1² + 2² + 3² +... + n²) / n³ and beta = lim(n->inf) [(1³ - 1²) + (2³ - 2²) +... + (n³ - n²)] / n⁴. Which of the following relations between alpha and beta is correct?
⚔️ Practice JEE Advanced Maths free + battle 1v1 →