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Let ℝ denote the set of all real numbers. Define the function f: ℝ → ℝ by f(x) = { 2 − 2x² − x² sin(1/x), if x ≠ 0 2, if x = 0. Then which one of the following statements is TRUE?

1. The function f is NOT differentiable at x = 0
2. There is a positive real number δ, such that f is a decreasing function on the interval (0, δ)
3. For any positive real number δ, the function f is NOT an increasing function on the interval (−δ, 0)
4. x = 0 is a point of local minima of f

Correct answer: There is a positive real number δ, such that f is a decreasing function on the interval (0, δ)

Solution

The given function f is defined piecewise, and by analyzing its behavior in the interval (0, δ) for a small positive δ, we can show that f is a decreasing function, making option B the correct statement.

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