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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