Differentiate x^3 − 3x^2 − 9x. Hence find the x-coordinates of the stationary points on the curve y = x^3 − 3x^2 − 9x
To differentiate, we bring the power down and decrease the power by 1. So x3 becomes 3x2, -3x2 becomes -6x, and -9x (which can be written as -9x1 ) becomes -9. So y' = 3x2 - 6x - 9 This equation tells us the gradient of the graph for any value of x, and we should be able to recall that at a stationary point, the gradient will be 0. We set y' to 0 and solve for x by factorising. 0 = 3x2 - 6x - 9 = (3x +3)(x - 3) So 3x + 3 = 0, hence x = -1 is a stationary point, and x - 3 = 0, hence x = 3 is a stationary point.
