<p><strong>725.</strong> If the curve \(f(x) = 3x^3 + ax^2 + bx\) where \(a, b\) are non negative integers, cuts the \(x\)-axis at 3 distinct points. Find the minimum value of \((a+b)\).</p>
Step-by-Step Solution
Key Concept: For a cubic f(x) = 3x³ + ax² + bx to have 3 distinct real roots, one root must be 0, and the quadratic factor must have 2 distinct non-zero real roots. This requires the discriminant condition a² > 12b combined with the constraint that the local extrema (found via f'(x) = 0) have opposite signs.
<p><strong>Step 1:</strong> Factor the function: f(x) = x(3x² + ax + b). Since f(x) cuts the x-axis at 3 distinct points, one root is x = 0, and 3x² + ax + b = 0 must yield 2 distinct non-zero real roots.</p><p><strong>Step 2:</strong> For 3x² + ax + b = 0 to have 2 distinct real roots: Δ = a² - 12b > 0, so a² > 12b.</p><p><strong>Step 3:</strong> For both roots to be non-zero, we need b ≠ 0. Since b is a non-negative integer, b ≥ 1.</p><p><strong>Step 4:</strong> For non-negative integers a and b with a² > 12b and b ≥ 1:<br/>• If b = 1: a² > 12 ⟹ a ≥ 4 (since a² ≥ 16). Minimum: a = 4, giving a + b = 5<br/>• If b = 2: a² > 24 ⟹ a ≥ 5. Minimum: a + b = 7<br/>• Higher values of b give larger sums.</p><p><strong>Step 5:</strong> Verify a = 4, b = 1: f(x) = x(3x² + 4x + 1) = x(3x + 1)(x + 1). Roots are 0, -1/3, -1 ✓ (3 distinct roots).</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: 5