Find the slopes of the tangent lines to the graph of the function f (x) = – 4 / x intersecting

Find the slopes of the tangent lines to the graph of the function f (x) = – 4 / x intersecting the line y = x at the point with the abscissa x = -1.

Consider the function f (x) = -4 / x. As you know, the equation of the tangent to the graph of the function f (x) at the point x0 has the form: y = f (x0) + f ‘(x0) * (x – x0). For our function f (x) = -4 / x, we get: f Ꞌ (x) = (-4 / x) Ꞌ = -4 * (x-1) Ꞌ = -4 * (-1) * x-1 – 1 = 4 / x² and f ‘(x0) = 4 / (x0) ². Therefore, the equation of the tangent to the graph of our function f (x) at the point x0 has the form: у = f (x0) + (4 / (x0) ²) * (x – x0) = -4 / x0 + (4 / (x0 ) ²) * x – 4 / x0 = (4 / (x0) ²) * x – 8 / x0, that is, y = (4 / (x0) ²) * x – 8 / x0.
According to the condition of the task, the tangent to the graph of this function intersects the straight line y = x at the point with the abscissa x = -1. It is clear that the ordinate of the intersection point also has the value y = x = -1. Therefore, the point of intersection with coordinates (–1; -1) must satisfy the tangent equation. Therefore, we have: -1 = (4 / (x0) ²) * (-1) – 8 / x0. This equation for x0 after appropriate transformations can be reduced to the form (x0) ² – 8 * x0 – 4 = 0 and has two different roots: (x0) 1 = 4 – 2√ (5) and (x0) 2 = 4 + 2√ (5).
Find the slope of the tangent for each root. When x0 = 4 – 2√ (5), the slope is 4 / (x0) ² = 4 / (4 – 2√ (5)) ² = 4 / (16 – 16√ (5) + 20) = 9 + 4√ (5). Similarly, if x0 = 4 – 2√ (5), then we have: 4 / (x0) ² = 4 / (4 + 2√ (5)) ² = 4 / (16 + 16√ (5) + 20) = 9 – 4√ (5).
Answer: 9 + 4√ (5) and 9 – 4√ (5).