<p><span class="math-inline">\(f(x)=\begin{cases}1-x & 0\le x\le 1\\ x+2 & 1<x<2\\ 4-x & 2\le x\le 4\end{cases}\)</span>. For <span class="math-inline">\(y=f(f(x))\)</span> on <span class="math-inline">\([0,4]\)</span>:</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Compute f(f(x)) on each sub-interval:</p><p>x∈[0,1]: f(x)=1-x∈[0,1]. f(f(x))=f(1-x)=1-(1-x)=x. Continuous, differentiable.</p><p>x∈(1,2): f(x)=x+2∈(3,4). f(f(x))=f(x+2)=4-(x+2)=2-x. Continuous, differentiable.</p><p>x∈[2,4]: f(x)=4-x∈[0,2]. If 4-x∈[0,1] (x∈[3,4]): f(f(x))=1-(4-x)=x-3. If 4-x∈(1,2) (x∈(2,3)): f(f(x))=(4-x)+2=6-x. At x=2: f(f(2))=f(2)=2, from left: 2-2=0 ≠ 2... actually from (1,2) branch at x=2⁻: f(f(x))→2-2=0. But x=2 is in [2,4]: f(f(2))=f(2)=f(4-2)=f(2)... loop. f(2)=4-2=2, f(f(2))=f(2)=2.</p><p>At x=1: from [0,1]: f(f(1))=1. From (1,2): f(f(1⁺))=2-1=1. Continuous (A) ✓.</p><p>f(f(x)): slope from [0,1] is 1, from (1,2) is -1. At x=1: LHD=1, RHD=-1. Not differentiable (B) ✓.</p><p>(C),(D): at x=2,3 check similarly — continuous and differentiable at 2, not differentiable at 3.</p><p><strong>Answer: (A),(B)</strong></p><div class="key-concept"><strong>Key Concept:</strong> Trace f(f(x)) through each interval carefully</div></div>
Correct Answer: A,B