# List of Notes - Category: Theorems of Differential Calculus

## 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.

## Rolle's Theorem

Rolle's Theorem. Suppose following three condition hold for function y=f(x):

1. function is defined and continuous on closed interval [a,b];
2. exists finite derivative f'(x) on interval (a,b);
3. f(a)=f(b).

then there exists point c (a<c<b) such that f'(c)=0.

## Mean Value Theorem

Mean Value Theorem (Lagrange Theorem). Suppose that function y=f(x) is defined and continuous on closed interval [a,b] and exists finite derivative f'(x) on interval (a,b). Then there exists point c (a<c<b) such that (f(b)-f(a))/(b-a)=f'(c).