Newton's method in optimization

In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x ) = 0 {\displaystyle f(x)=0} . However, to optimize a twice-differentiable f {\displaystyle f} , our goal is to find the roots of f ′ {\displaystyle f'} . We can therefore use Newton's method on its derivative f ′ {\displaystyle f'} to find solutions to f ′ ( x ) = 0 {\displaystyle f'(x)=0} , also known as the critical points of f {\displaystyle f} . These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function f {\displaystyle f} .