Quadratic Equations
Quadratic function constraints
Grade 11
Question:
<p>Let function \(f(x)\) be defined as \(f(x) = x^2 + bx + c\), where \(b\), \(c\) are real numbers and \(f(1) - 2f(5) + f(9) = 32\). Ordered pairs \((b, c)\) such that \(|f(x)| \leq 8\) for all \(x\) in the interval \([1, 9]\), then</p>
<p>(a) \(b = 7\)</p>
<p>(b) \(b = -10\)</p>
<p>(c) \(c = 17\)</p>
<p>(d) \(b + c = 24\)</p>
Step-by-Step Solution
Key Concept: Use the constraint f(1) - 2f(5) + f(9) = 32 to establish a relationship involving the coefficient b, then recognize that the second difference of a quadratic equals 2a(spacing)². Apply the condition |f(x)| ≤ 8 on [1,9] by analyzing the extrema of f at critical points and endpoints.
<p><strong>Step 1: Find coefficient a using second differences</strong></p><p>For f(x) = x² + bx + c, the second difference is:</p><p>f(1) - 2f(5) + f(9) = (1 + b + c) - 2(25 + 5b + c) + (81 + 9b + c)</p><p>= 1 + b + c - 50 - 10b - 2c + 81 + 9b + c = 32</p><p>This gives: f(1) - 2f(5) + f(9) = 32 ✓ (confirms a = 1)</p><p><strong>Step 2: Apply constraint at key points</strong></p><p>We need |f(x)| ≤ 8 for all x ∈ [1,9]:</p><p>• f(1) = 1 + b + c: |1 + b + c| ≤ 8 → -9 ≤ b + c ≤ 7</p><p>• f(5) = 25 + 5b + c: |25 + 5b + c| ≤ 8 → -33 ≤ 5b + c ≤ -17</p><p>• f(9) = 81 + 9b + c: |81 + 9b + c| ≤ 8 → -89 ≤ 9b + c ≤ -73</p><p><strong>Step 3: Solve the system</strong></p><p>From the three inequalities, the vertex occurs at x = -b/2. Since b < 0 (from the constraints), check if vertex lies in [1,9].</p><p>Subtracting equations systematically:</p><p>From 5b + c ≥ -33 and b + c ≤ 7: 4b ≥ -40 → b ≥ -10</p><p>From 9b + c ≥ -89 and 5b + c ≤ -17: 4b ≥ -72 → b ≥ -18</p><p>From 9b + c ≤ -73 and b + c ≥ -9: 8b ≤ -64 → b ≤ -8</p><p><strong>Step 4: Find exact value</strong></p><p>The binding constraints give b = -8, and substituting back: c = 15</p><p>∴ Answer: B (The ordered pair (b,c) = (-8, 15))
Correct Answer: B