# Illustrating Simple Variation Problem

Calculus of variations is a subject where you study the extrema (maximum or minimum) of a function of functions.

We know that, to find the extrema (maximum or minimum) of a “functional” of the form

$v[f(x)]=\int_a^b F(x,f(x),f'(x))dx; f(a)=f_1,f(b)=f_2$

we first identify and substitute $F(x,f,f')$ in Euler Equation (Or sometimes called Euler-Lagrange Equation)

${\partial F\over\partial f} - {d\over dx} {\partial F\over\partial f'} =0$

to obtain a Ordinary Differential Equation, solving which gives the extrema we are looking for, making use of the boundary conditions.

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