Prove directly that the functions f1x1 f2xx and f3xx2 are li

Prove directly that the functions

f₁(x)=1, f₂(x)=x and f₃(x)=x²

are linearly independent on the whole real line.

(Assume that C₁=C₂=C₃=0. Differentiate this equation twice, and conclude from the equations you get that C₁=C₂=C₃=0)

Solution

Let any point on the number line be of the form

c1f1(x) + c2f2(x) +c3f3(x)

= c1+c2x+c3x^2

Let F(x) = c1+c2x+c3x^2

F\'(x) = c2+2c3x

F\"(X) = 2c3

As F\"(X)>0, F has a minimum at x = -c2/2c3

Hence c3 cannot be 0 as x is not defined

So all c1, c2, c3 cannot be equal to 0 at the same time.

So f1(x), f2(x) f3(x) are linearly independent.