# Fermat's Theorem

Knowing derivative of function f(x) allows us to make some conclusions about behavior of function y=f(x).

Fermat's Theorem. Suppose function y=f(x) is defined on interval [a,b] and at some point c in (a,b) takes maximum (minimum) value. If exist finite derivative f'(c) then f'(c)=0.

Note, that converse of Fermat's Theorem is not true, that is, if f'(c)=0 then f doesn't necessarily have maximum or minimum at c.

For example, consider function f(x)=x^3 then f'(x)=3x^2 and f'(x)=0 when x=0, but x=0 is neither maximum nor minimum. It is just a point where tangent line is horizontal and intersects graph.

Note that there may be maximum (or minimum) where f'(c) does not exist. For instance, the function f(x)=|x| has its minimum value at 0, but the value cannot be found by setting f'(x)=0 because f'(0) does not exist.
Geometric interpretation of Fermat's theorem is fairly simple. Theorem states that if at point c there is maximum or minimum then tangent line at that point is horizontal, i.e. its slope equals 0.

Also, we require point c to be inner point of interval [a,b]. If point c is endpoint (either a or b) then Fermat's theorem doesn't hold.