phukraut

how to find a min/max of f(x):

take the first derivative of f(x), call it f'(x). find all the points where f'=0. plot them on a number line for x. pick a number between these points, and plug that number into f'(x) and see if the result is greater than or less than zero. do that for all the intervals between your "zero"-points. this will tell you where the curve goes up and down.

example: f(x)=x^2. f'(x)=2x. f'(0)=0, so this is a critical point. try f'(-10)=-20<0. f'(10)=20>0.


f'<0 here ||| f'>0 here
---------0-----------.

if a curve goes up (f' greater than zero), then reaches a zero point (f'=0), then goes down (f' less than zero), you have a local maximum. if it goes down, then zero, then up, then you have a local minimum.

to find out what the actual max/min point is, you plug the zero-point into f(x) (the original function), and get a y coordinate. for example, if f'(M)=0 is your only zero point, and f'(M-1)>0 and f'(M+1)<0, then you have a local maximum at the point (M, f(M)).

this process is caleld the First Derivative Test.

points of inflection have to do with how the curve goes up or down. is it concave up or concave down? for example, does it go

\___/, (concave up)

or does it go
___
/ \? (concave down)

in order to find out, you take the second derivative, and find out where it is zero. if f''>0, then f is concave up. if f''<0, then f is concave down. you can also find min/max via concavity. if f'=0 at an extreme point, and f''>0, then f has a local min at this point. if f'=0 and f''<0, then f has a local max. this is called the Second Derivative Test. an inflection point is where the curve goes from concave up to concave down or vice versa. to find the point of inflection, determine where the concavity changes in x, and plug that value into f(x) to get the y coordinate.