Limits, Continuity & Differentiability
Differentiability
Grade None
Question:
<p>If <span>\(y = f(x)\)</span> is differentiable for all <span>\(x \in \mathbb{R}\)</span>, then:</p>
<p>(a) <span>\(y = |f(x)|\)</span> is differentiable for all <span>\(x \in \mathbb{R}\)</span></p>
<p>(b) <span>\(y = f^2(x)\)</span> is non-differentiable for at least one <span>\(x\)</span></p>
<p>(c) <span>\(y = f(x)|f(x)|\)</span> is non-differentiable for at least one <span>\(x\)</span></p>
<p>(d) <span>\(y = |f(x)|^3\)</span> is differentiable for all <span>\(x \in \mathbb{R}\)</span></p>
Step-by-Step Solution
Key Concept: Absolute value and composite functions may introduce non-differentiability at points where the inner function equals zero, but higher odd powers smooth out the derivative.
<p>For option (d), <span>$y = |f(x)|^3 = (|f(x)|)^3$</span>. Since <span>$|f(x)|^3$</span> is always differentiable (the cube of any continuous function is differentiable), this statement is true.</p>
Correct Answer: D