How do I find the co-ordinates of a stationary point of a given line and determine whether it is a minimum or a maximum point?
When you find the first derivative of the equation and equate it to 0 then solve for x, you will find the x-coordinates of the stationary point(s) of the graph. For example, let's say f(x)=-x3+x2+8x+10 then f'(x)=-3x2+2x+8. Hence, equating that to 0 will lead to x=2 and x=-4/3.For these x-coordinates, plotting them back into f(x) will give you the y-coordinate. So, the coordinates of the stationary points would be (2,22) and (-4/3,94/27).Now, to determine whether these are maximum or minimum points without drawing out the graph sketch, you will need to find the second derivative, in this case f''(x)=-6x+2. Inputting the same x-coordinates into f''(x) will show you the acceleration of the graph at those points: a positive number means it's a minimum point, a negative number means a maximum point and if f''(x) equals 0 in some rare cases, it means it is a point of inflection.Hence, f''(2)=-10<0 so the stationary point (2,22) is a maximum point.f''(-4/3)= 10>0 so the stationary point (-4/3,94/27) is a minimum point.
