Final answer:
The tangent line to the function f(x) = x³ - 3x is horizontal where the derivative is zero. By finding the derivative and setting it to zero, we determine the x-values where the horizontal tangents occur, which are x = 1 and x = -1. The corresponding points on the function are (1, -2) and (-1, 2).
Step-by-step explanation:
The goal is to find the points where the tangent line to the function f(x) = x³ - 3x is horizontal. A horizontal tangent line occurs where the derivative of the function is zero, since the slope of the tangent line at any point is given by the derivative of the function at that point.
To find these points, we first calculate the derivative of f(x), which is f'(x) = 3x² - 3. Setting this equal to zero gives us the equation 3x² - 3 = 0, which simplifies to x² = 1. This has two solutions: x = 1 and x = -1.
Substituting these x values back into the original function, we find that f(1) = 1³ - 3(1) = -2 and f(-1) = (-1)³ - 3(-1) = 2. Therefore, the points where the tangent line is horizontal are (1, -2) and (-1, 2).