Fermat's Theorem (stationary Points) - Intuition

Intuition

Intuitively, a differentiable function is approximated by its derivative – a differentiable function behaves infinitesimally like a linear function or more precisely, Thus, from the perspective that "if f is differentiable and has non-vanishing derivative at then it does not attain an extremum at " the intuition is that if the derivative at is positive, the function is increasing near while if the derivative is negative, the function is decreasing near In both cases, it cannot attain a maximum or minimum, because its value is changing. It can only attain a maximum or minimum if it "stops" – if the derivative vanishes (or if it is not differentiable, or if one runs into the boundary and cannot continue). However, making "behaves like a linear function" precise requires careful analytic proof.

More precisely, the intuition can be stated as: if the derivative is positive, there is some point to the right of where f is greater, and some point to the left of where f is less, and thus f attains neither a maximum nor a minimum at 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".

The intuition is based on the behavior of polynomial functions. Assume that function f has a maximum at x₀, the reasoning being similar for a function minimum. If is a local maximum then, roughly, there is a (possibly small) neighborhood of such as the function "is increasing before" and "decreasing after" . As the derivative is positive for an increasing function and negative for a decreasing function, is positive before and negative after . doesn't skip values (by Darboux's theorem), so it has to be zero at some point between the positive and negative values. The only point in the neighbourhood where it is possible to have is .

The theorem (and its proof below) is more general than the intuition in that it doesn't require the function to be differentiable over a neighbourhood around . It is sufficient for the function to be differentiable only in the extreme point.