Find the stationary points of the function f(x) = x^3+6x^2+2 and determine if they are local maximums or minimums.

A stationary point is defined as a point on the function where the gradient is zero, i.e f'(x) = 0. To construct this equation we must differentiate the function, we would find f'(x) = 3x²+12x. The corresponding equation is then 3x²+12x = 0, which we solve for the x co-ordinates of any stationary points of the function. We find x₁= 0 and x₂=-4, to find the y co-ordinates of each point we simply substitute x₁ and x₂ into f(x), we find that points P₁ = (0,2) and P₂ = (-4,34) are stationary.To determine whether these points are local maxima or minima we must identify the the change in the gradient around the point is positive or negative. To do this we must differentiate the function again, finding that f''(x) = 6x+12. By substituting in the x co-ordinate of each stationary point we find that f''(x₁) = 12 and f''(x₂) = -12. This tells us that around P₁ the gradient increases either side of the point, indicating it is a local minimum. Conversely we identify that P₂ is a local maximum.