Sets, Relations & Functions
Grade 11
Question:
<p>For \(f(x)=\max\{x,x^2,x^3,x^4\}\). Which are correct?<br>(A) \(f(x)=x^4\) for \(x\le -1\) (B) \(f(x)=x^2\) for \(-1<x\le 0\) (C) \(f(\frac{1}{2})=\frac{1}{2}\) (D) \(f(\frac{1}{2})=\frac{1}{4}\)</p>
A,B,C,D
<strong>A,B,C</strong>
A,C,D
B,C
Step-by-Step Solution
<div class="solution"><p>(A) x\le-1: x^4\gex^2\ge0 while x,x^3<0 \to max=x^4 ✓</p><p>(B) -1<x\le0: x^2\gex^4, x^2\gex, x^2\gex^3 \to max=x^2 ✓</p><p>(C) Values at x=1/2: 1/2,1/4,1/8,1/16 \to max=1/2 ✓; (D) false.</p><p><strong>Answer: (A),(B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> On (0,1), higher powers are smaller, not larger.<div class="key-concept"><strong>Key Concept:</strong> max of powers -- interval splitting by sign and magnitude
Correct Answer: A,B,C