Sets, Relations & Functions
General
Grade 11
Question:
<p><span class="math-inline">\(f:[0,3]\to[1,29]\)</span>, <span class="math-inline">\(f(x)=2x^3-15x^2+36x+1\)</span>. Nature of <span class="math-inline">\(f\)</span>?</p>
One-one and onto
Onto but not one-one
One-one but not onto
Neither
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f'(x)=6(x-2)(x-3): not monotone → not one-one. f(0)=1, f(2)=29, f(3)=28 → range=[1,29] → onto.</p><p><strong>Answer: (B) onto but not one-one</strong></p><div class="trap-box"><strong>Trap:</strong> A cubic on a closed interval is not automatically one-one.</div><div class="key-concept"><strong>Key Concept:</strong> Derivative sign + extreme values determines mapping nature on closed interval</div></div>
Correct Answer: 2