Final answer:
To find the points on the curve where the tangent is horizontal, we must set the derivative of the function, 3x^2 - 6x - 9, to zero and solve for x. By doing this, we find the x-values where the slope of the tangent is zero, which is where the tangent to the curve is horizontal.
Step-by-step explanation:
To find the points on the curve y = x³ - 3x² - 9x + 1 where the tangent is horizontal, we need to determine where the slope of the tangent to the curve is equal to zero. The slope of the tangent line at any point on the curve is given by the derivative of the function. In this case, the derivative of y with respect to x is y' = 3x² - 6x - 9.
To find the points where the tangent is horizontal, we set the derivative equal to zero:
3x² - 6x - 9 = 0
By solving this quadratic equation, we can find the values of x for which the tangent is horizontal. We can then plug these x-values back into the original equation for y to obtain our points.