<p>What is the maximum height of any point on the curve \(y = -x^2 + 6x - 5\) above the \(x\)-axis?</p>
Step-by-Step Solution
Key Concept: The maximum height above the x-axis occurs at the vertex of the parabola. Since the parabola opens downward (negative leading coefficient), the y-coordinate of the vertex gives the maximum height.
<p><strong>Step 1:</strong> Identify that for y = -x² + 6x - 5, the parabola opens downward (a = -1 < 0), so maximum occurs at vertex.</p><p><strong>Step 2:</strong> Find vertex using x-coordinate: x = -b/(2a) = -6/(2·(-1)) = -6/(-2) = 3</p><p><strong>Step 3:</strong> Substitute x = 3 into the equation to find maximum y-value:</p><p>y = -(3)² + 6(3) - 5 = -9 + 18 - 5 = 4</p><p><strong>Step 4:</strong> Verify this is above x-axis (y = 4 > 0) ✓</p><p>∴ Maximum height above x-axis = <strong>4</strong></p>
Correct Answer: 4