Applications of Derivatives
Local Extrema Conditions
Grade 12
Question:
<p>Let <i>S</i> be the set of real values of parameter <i>λ</i> for which the equation \(f(x) = 2x^3 - 3(2 + \lambda)x^2 + 12\lambda x\) has exactly one local maximum and exactly one local minimum. Then, <i>S</i> is a subset of</p>
<p>(a) \((-4, 1)\)</p>
<p>(b) \((-3, 3)\)</p>
<p>(c) \((3, \infty)\)</p>
<p>(d) \((-1, 0)\)</p>
Step-by-Step Solution
Key Concept: A cubic has exactly one local max and min when its derivative (quadratic) has two distinct real roots
<p><strong>Step 1:</strong> Find $f'(x) = 6x^2 - 6(2 + \lambda)x + 12\lambda$</p><p><strong>Step 2:</strong> For exactly one local max and one local min, $f'(x) = 0$ must have two distinct real roots.</p><p><strong>Step 3:</strong> Discriminant $\Delta = 36(2 + \lambda)^2 - 4(6)(12\lambda) > 0$</p><p><strong>Step 4:</strong> Simplify: $(2 + \lambda)^2 - 8\lambda > 0 \Rightarrow \lambda^2 - 4\lambda + 4 > 0 \Rightarrow (\lambda - 2)^2 > 0$</p><p><strong>Step 5:</strong> This holds for all $\lambda \neq 2$, which is within $(-3, 3)$.</p><p>∴ Answer is B.</p>
Correct Answer: B