please answer with steps Horizontal tangents Find equations for the horizontal t

Question

please answer with steps

Horizontal tangents Find equations for the horizontal tangents to the curve y = x3 - 3x - 2. Also find equations for the lines that arc perpendicular to these tangents at the points of tangency. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope? Find an equation for the line that is perpendicular to the curve's tangent at this point.

Explanation / Answer

y = x^3-3x-2 a)Horizontal tangent => m = 0 m = dy/dx = 3x^2 - 3 = 0 =>3x^2 = 3 =>x^2 = 1 =>x = 1, -1 =>y = -4, 0 So equation of horizontal tangents are, y = 0, y+4 = 0 Equations of lines perpendicular to these tangents at the points of tangentcy are: x+1 = 0 and x-1 = 0 b)m = 3x^2-3 Smallest slope = -3 Point (x,y) = (0,-2) Slope of line perpendicular to tangent = 1/3 So equation is y+2 = (1/3)x =>3y+6 = x =>x-3y = 6

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