List of Notes - Category: Theorems of Differential Calculus

Fermat's Theorem

Knowing derivative of function `f(x)` allows us to make some conclusions about behaviour 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)`.