Let f(x)= x^3 -9x^2 -81x + 12. Calculate f'(x) and f''(x). Use f'(x) to calculate the x-values of the stationary points of this function.

To answer this question we need to first decide what the question is asking for. In this case the question is asking for the first and second differential of a given function. If we have a function f(x) = xⁿ then the general formula for the differential is f'(x) = nx^n-1. Also, as a rule, any constant term in the function will differentiate to zero. The second differential can be calculated by differentiating the first differential.
We can use these rules to answer the question. f'(x) = 3x² -18x -81 f''(x) = 6x -18
The question then asks us to find the stationary points . To calculate the stationary points of the function, we must solve f'(x) = 0. i.e. 3x² -18x -81 = 0. This can be simplified to x² -6x -27 = 0.We then factorise to get (x-9)(x+3) = 0. Therefore we find the stationary points are at x = 9 or -3