Find and classify all the stationary points of the function f(x) = x^3 - 3x^2 + 8
Stationary points occur when the gradient of the function is zero. They can be visualised on a graph as hills (maximum points), as troughs (minimum points), or as points of inflection. (I would draw all three examples on the screen). Firstly, we must find the first derivative and set it equal to zero because this is the gradient function. This gives us 3x^2 – 6x = 0. Factorising gives us 3x(x – 2) = 0 meaning that either 3x = 0 or x – 2 = 0. Therefore the stationary points are at x = 0 and x = 2. To classify the points, we must find the second derivative of the function and sub in the x values of the stationary points. If the value of the second derivative is greater than 0 it is a minimum point, if it is less than 0, then it is a maximum point. For this function, the second derivative is 6x – 6. For the point at x = 0, the second derivative is -6, making it a maximum point. For the point at x = 2, the second derivative is 6, making it a minimum point.
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