Second Derivatives and Beyond
We say x = c is a critical point of the function f if f (c) exists and f'(c) = 0 or is undefined. It's generally a peak or valley in the curve. It's where the slopes becomes interesting. When climbing Mount Everest, we might say that we've reached the critical point when we've reached the summit. It's where we can enjoy the view while feasting on granola.
Be Careful:A critical point of a function f is a value in the domain of f at which the derivative f ' is 0 or undefined.
It's also possible for a function to have no critical points at all.
To find the critical points of a function f we
- take the derivative of f
- find any places where f ' is undefined but f is defined, and finally
- find the roots of the derivative.
We can often check our answers by graphing the function and making sure it looks like it has critical points in the right places. We know what a graph looks like at a spot where the derivative doesn't exist:
and we know what a graph looks like when it has a derivative of 0 (horizontal tangent line):