Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( f(x) = \max\{x, x^3\ \), \( x \in \mathbb{R} \), where \( \mathbb{R} \) is the set of all real numbers. The set of all points where \( f(x) \) is NOT differentiable is:}

• A. \( \{-1, 1, 2\} \)
• B. \( \{-2, -1, 1\} \)
• C. \( \{0, 1\} \)
• D. \( \{-1, 0, 1\} \)

Hint:

To find points of non-differentiability for a piecewise-defined function, analyze where the two components intersect or where the derivative changes abruptly.

Answer 1

The Correct Option is D

Step 1: Analyze the function \( f(x) \). The function is defined as: \[ f(x) = \max\{x, x^3\}. \] This implies:
For \( x \geq 1 \), \( f(x) = x \), as \( x>x^3 \).
For \( -1 \leq x \leq 1 \), \( f(x) = x^3 \), as \( x^3>x \).
For \( x \leq -1 \), \( f(x) = x \), as \( x>x^3 \). Step 2: Points of non-differentiability. The function \( f(x) \) is not differentiable at the points where the two components \( x \) and \( x^3 \) intersect. Solve \( x = x^3 \): \[ x^3 - x = 0 \implies x(x^2 - 1) = 0 \implies x = 0, \, x = 1, \, x = -1. \] Final Answer: \[ \boxed{\{-1, 0, 1\}} \]