Sets, Relations & Functions
General
Grade None
Question:
<p>\(f:[0,3]\to[1,29]\), \(f(x)=2x^3-15x^2+36x+1\). Nature of \(f\)?</p>
One-one and onto
Onto but not one-one
One-one but not onto
Neither
Step-by-Step Solution
<div class="solution"><p>f'(x)=6(x-2)(x-3): not monotone \to not one-one. f(0)=1, f(2)=29, f(3)=28 \to range=[1,29] \to 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 class="key-concept"><strong>Key Concept:</strong> Derivative sign + extreme values determines mapping nature on closed interval
Correct Answer: 2