Sets, Relations & Functions
General
Grade 11
Question:
<p>For <span class="math-inline">\(f(x)=\max\{x,x^2,x^3,x^4\}\)</span>. Which are correct?<br>(A) <span class="math-inline">\(f(x)=x^4\)</span> for <span class="math-inline">\(x\le -1\)</span> (B) <span class="math-inline">\(f(x)=x^2\)</span> for <span class="math-inline">\(-1<x\le 0\)</span> (C) <span class="math-inline">\(f(\frac{1}{2})=\frac{1}{2}\)</span> (D) <span class="math-inline">\(f(\frac{1}{2})=\frac{1}{4}\)</span></p>
A,B,C,D
<strong>A,B,C</strong>
A,C,D
B,C
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(A) x≤-1: x⁴≥x²≥0 while x,x³<0 → max=x⁴ ✓</p><p>(B) -1<x≤0: x²≥x⁴, x²≥x, x²≥x³ → max=x² ✓</p><p>(C) Values at x=1/2: 1/2,1/4,1/8,1/16 → 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><div class="key-concept"><strong>Key Concept:</strong> max of powers — interval splitting by sign and magnitude</div></div>
Correct Answer: A,B,C