How Do You Prove A Derivative Of A Function Is Zero At Its Supremum?
It’s very intuitive. In order to be at it’s greatest value, the function either has to be constant (if it’s constant the derivative is 0 anyway) or RISE there (at a positive derivative). When it stops rising, the derivative changes to 0. Generally, at any extremum (global or local) the derivative of a function is 0, as visually the extremum is either a peak or a valley. In order to produce such a shape, it has to first fall, then rise, or first rise then fall. This way, the derivative has to change signs, and this can only be done by passing 0.
