GATE XE (Engineering Sciences) 2024 - Question 14

Section: Engineering Mathematics | Marks: 1

Assume that 𝑓: [0, 1] →ℝ is continuous on [0, 1] and differentiable on (0, 1) such that 𝑓(𝑥+ ℎ) = 𝑓(𝑥) + ℎ𝑓′(𝑥+ 𝜃ℎ) for some 0 < 𝜃< 1. If 𝑓(𝑥) = 𝑥2(1 + 𝑥), and 𝜃 is expressed in terms of 𝑥 and ℎ, then the value of lim ℎ→0 𝜃(𝑥, ℎ) is

• A. 1 3
• B. 1 2
• C. 1 4
• D. 4 5

Answer: B. 1 2