<p>The area (in sq. units) of the region described by \(A = \{(x, y)\mid y \geq x^2 - 5x + 4,\ x + y \geq 1,\ y \leq 0\}\) is</p>
Step-by-Step Solution
Key Concept: Identify the three boundary curves, find their intersection points, and determine which region satisfies all three inequalities simultaneously by checking test points.
<p><strong>Step 1:</strong> Identify the boundary curves:</p><ul><li>Parabola: y = x² - 5x + 4 = (x-1)(x-4)</li><li>Line: x + y = 1, or y = 1 - x</li><li>Horizontal line: y = 0</li></ul><p><strong>Step 2:</strong> Find key intersections.</p><ul><li>Parabola meets x-axis: y = 0 ⟹ x² - 5x + 4 = 0 ⟹ x = 1 or x = 4</li><li>Line y = 1 - x meets parabola: 1 - x = x² - 5x + 4 ⟹ x² - 4x + 3 = 0 ⟹ x = 1 or x = 3</li><li>At x = 1: parabola gives y = 0; line gives y = 0. Point (1, 0)</li><li>At x = 3: parabola gives y = 9 - 15 + 4 = -2; line gives y = -2. Point (3, -2)</li></ul><p><strong>Step 3:</strong> Determine the valid region. The region must satisfy:</p><ul><li>y ≥ x² - 5x + 4 (above or on parabola)</li><li>x + y ≥ 1 (above or on line)</li><li>y ≤ 0 (on or below x-axis)</li></ul><p>For 1 ≤ x ≤ 3: the binding constraints are y = 1 - x (upper bound) and y = x² - 5x + 4 (lower bound), with both ≤ 0 in this interval.</p><p><strong>Step 4:</strong> Calculate the area:</p><p>A = ∫₁³ [(1 - x) - (x² - 5x + 4)] dx = ∫₁³ [1 - x - x² + 5x - 4] dx</p><p>= ∫₁³ [-x² + 4x - 3] dx = [−x³/3 + 2x² − 3x]₁³</p><p>= [−9 + 18 − 9] − [−1/3 + 2 − 3] = 0 − (−4/3) = <strong>4/3</strong></p><p>∴ Answer: B</p>
Correct Answer: B