<p>The equation of the tangent to the curve \(y = x - \frac{8}{x^2}\) which is parallel to the <i>x</i>-axis is</p>
Step-by-Step Solution
Key Concept: A tangent parallel to the x-axis has slope 0, so we find where dy/dx = 0, then determine the y-coordinate at that point to get the tangent line equation.
<p><strong>Step 1:</strong> Find the derivative of y = x - 8/x²</p><p>dy/dx = 1 - 8·(-2x)/x⁴ = 1 + 16/x³</p><p><strong>Step 2:</strong> Set dy/dx = 0 for tangent parallel to x-axis</p><p>1 + 16/x³ = 0</p><p>16/x³ = -1</p><p>x³ = -16</p><p>x = -2</p><p><strong>Step 3:</strong> Find y-coordinate by substituting x = -2 into the original curve</p><p>y = (-2) - 8/(-2)² = -2 - 8/4 = -2 - 2 = -4</p><p><strong>Step 4:</strong> Write the equation of tangent line (horizontal line through (-2, -4))</p><p>The tangent line is y = -4</p><p>∴ Answer: C</p>
Correct Answer: C