how do you find and define the nature of a point of inflextion, i am aware that setting the second derivative equal to zero finds them but how do you determine thier nature??
an elementary way to do it is look at the 2nd derivative just to the left and just to the right of the point of inflection. If it goes, from left to right, negative to positive then you know it goes from concave down to concave up. If it goes from pos to neg then you know it changes from conc. up to conc. down, etc.
If the value of the second derivative is less than 0...the point is a maxiumum point.
If the value is more than 0, it is a minimum point.
To find out if it is an inflective point, you take the x value either side of the x value you are checking and find its gradient.
For example if your checking the point where x = 0.
You find the gradient at x = -1 and x = 1
If value of the second derivative goes positive to negative...its an inflective point.
that site may explain more.
the inflection point is usually the zeros on the graph of the second derivative.so you can graph your equation and then find both the first and second derivative, the zeros will be the point of inflection. you can also solve for it by finding the second derivative of your equation and then finding the critical values then plugging into your data and finding where it goes from negative to positive as minimun and positive to negative as maximum.