Determine the coordinates of all the stationary points of the function f(x) = (1/3)*x^3+x^2-3*x+1 and state whether they are a maximum or a minimum.

To find the answer you must first differentiate the function and set this equal to zero. This forms the quadratic equation x^2+2x-3=0 which can then be solved either by factorisation or by using the quadratic formula. This enables you to find the x-coordinates of the two stationary points which can then be substituted back into the original equation to find the y-coordinates of the stationary points. The coordinates of the stationary points are (1, -2/3) and (-3, -8).To find the nature of the stationary points you must find the second differential of the original function which is f”(x)=2x+2. Then you substitute the x-coordinates into this function and if f”(x)<0 the point is a maximum, if f”(x)>0 then the point is a minimum. Therefore, we can determine that (-3, -8) is a maximum and (1, -2/3) is a minimum.