Relations & Functions
One-One Function
Grade 12
Question:
<p>Let <m>f : \mathbb{R} \to \mathbb{R}</m> defined by <m>f(x) = x^3 + ax^2 + 3x + 100</m>, then the values of <m>a</m> for which <m>f</m> is a one-one function, is</p>
<p>(a) <m>-2 \leq a \leq 3</m></p>
<p>(b) <m>-2 \leq a \leq 2</m></p>
<p>(c) <m>-3 \leq a \leq 3</m></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: A function is one-one if its derivative doesn't change sign, which requires the discriminant of f'(x) to be non-positive.
<p><strong>Solution:</strong></p><p>Given <m>f(x) = x^3 + ax^2 + 3x + 100</m></p><p>Taking derivative: <m>f'(x) = 3x^2 + 2ax + 3</m></p><p>For <m>f(x)</m> to be one-one, we need <m>f'(x) \geq 0</m> or <m>f'(x) \leq 0</m> for all <m>x</m>.</p><p>Since <m>f'(x)</m> is a quadratic expression with coefficient of <m>x^2 > 0</m>, for <m>f'(x) \geq 0</m>, the discriminant must be non-positive:</p><p><m>\Delta \leq 0</m></p><p><m>4a^2 - 36 \leq 0</m></p><p><m>a^2 \leq 9</m></p><p><m>-3 \leq a \leq 3</m></p><p>∴ Answer is (c).</p>
Correct Answer: C