More precisely, the intuition can be stated as: if the derivative is positive, there is "some point" to the right of formula_13 where "f" is greater, and "some point" to the left of formula_13 where "f" is less, and thus "f" attains neither a maximum nor a minimum at formula_29 Conversely, if the derivative is negative, there is a point to the right which is lesser, and a point to the left which is greater. Stated this way, the proof is just translating this into equations and verifying "how much greater or less".