<p>Let \(l\) be the line through \((0, 0)\) and tangent to the curve \(y = x^3 + x + 16\). Then the slope of \(l\) is equal to:</p>
Step-by-Step Solution
Key Concept: A line through the origin with slope m is tangent to the curve if it touches the curve at exactly one point, meaning both the point lies on the curve and the slope at that point equals m. This gives us two conditions to find the tangency point and slope.
<p><strong>Step 1:</strong> Let the tangent line have slope m and pass through origin (0,0), so its equation is: <br/>y = mx</p><p><strong>Step 2:</strong> This line is tangent to the curve y = x³ + x + 16 at some point (a, a³ + a + 16). At the point of tangency, two conditions must hold:<br/>(i) The point lies on the tangent line: a³ + a + 16 = ma<br/>(ii) The slope of the curve equals the slope of the line: dy/dx|ₓ₌ₐ = m</p><p><strong>Step 3:</strong> Find the derivative of the curve:<br/>dy/dx = 3x² + 1<br/>At x = a: 3a² + 1 = m</p><p><strong>Step 4:</strong> Substitute m = 3a² + 1 into condition (i):<br/>a³ + a + 16 = (3a² + 1)·a<br/>a³ + a + 16 = 3a³ + a<br/>16 = 3a³ - a³<br/>16 = 2a³<br/>a³ = 8<br/>a = 2</p><p><strong>Step 5:</strong> Find the slope m using m = 3a² + 1:<br/>m = 3(2)² + 1<br/>m = 3(4) + 1<br/>m = 12 + 1<br/>m = 13</p><p><strong>Step 6:</strong> Verify: The tangent line y = 13x at point (2, 26) gives 13(2) = 26, and point (2, 2³ + 2 + 16) = (2, 26) ✓<br/>∴ Answer: d</p>
Correct Answer: d