Fermat's theorem (stationary points)

In calculus, Fermat's theorem states that if a function has a local maximum or minimum at some point, then the derivative at that point must be zero if it exists.

 Proof 
Suppose  is a function that has a local maximum at . 

The definition of a local maximum says there is some range  such that  if . From this, we know that when ,

and when ,

The definition of the derivative says

By the above, these difference quotients can only be equal if they are both 0. This is Fermat's theorem.

If  has a minimum at , then  has a maximum at . Then by the above,

so the derivative of  must also be 0.

Category:Calculus