Let denote the set of all real numbers

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

\mathbb{R}

denote the set of all real numbers. Define the function

f: \mathbb{R} \rightarrow \mathbb{R}

f(x) = \begin{cases} 2 - 2x^2 - x^2 \sin \frac{1}{x} & \text{if } x \neq 0, \\ 2 & \text{if } x = 0. \end{cases}

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

\delta

, such that

f

is a decreasing function on the interval

(0, \delta)

\delta

, the function

f

is NOT an increasing function on the interval

(-\delta, 0)

(D)

x = 0

is a point of local minima of

f