A curve has equation y = x^3 - 3x^2 -24x + 5, find the x co-ordinates of the two stationary points of the curve and hence determine whether they are maximum or minimum points.
y = x3 - 3x2 - 24x + 5, First, calculate the derivative of y and find its roots when y = 0:dy/dx = 3x2 - 6x -24 = 0 -> x2 - 2x - 8 = 0 -> (x+2)(x-4) = 0Therefore the coordinates of the stationary points are x = -2, 4. Now calculate the second derivative of y and insert these x values:d2y/dx2 = 6x - 6,For x = -2: d2y/dx2 = -12 - 8 = -18, this result is < 0 so this point is a maximum point.For x = 4, d2y/dx2 = 24 - 6 = 18, this result is > 0 so this point is a minimum point.
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