Let ℝ denote the set of all real numbers. Define the function f : ℝ → ℝ by f(x) = cases 2 - 2x^2 - x^2 1 x & if x 0, 2 & if x = 0. cases Then which one of the following statements is TRUE?

JEE Advanced 2025 Mathematics question

Let ℝ denote the set of all real numbers. Define the function f : ℝ → ℝ by

f(x) =

Then which one of the following statements is TRUE?

(A)The function f is NOT differentiable at x = 0
(B)There is a positive real number δ, such that f is a decreasing function on the interval (0, δ)
(C)For any positive real number δ, the function f is NOT an increasing function on the interval (−δ, 0)Correct
(D)x = 0 is a point of local minima of f

Answer: (C)

Solution

(A) , so f is differentiable at 0: false. For x ≠ 0, ; near 0 the term takes the values ±1 arbitrarily close to 0 on both sides while the other terms tend to 0, so f′

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Question ID VK-037790