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:}

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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: $$ \boxed{\{-1, 0, 1\}} $$

Answer

$ \{-1, 1, 2\} $

Answer

$ \{-2, -1, 1\} $

Answer

$ \{0, 1\} $

Answer

$ \{-1, 0, 1\} $

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:}

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The Correct Option is D

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