Are the functions f, g, and h given below linearly independent? f(x)=e^4x
g(x)=xe^4x
h(x)=x2e^4x.

If they are independent, write all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer.

A(e^4x−cos5x)+B(e^4x+cos5x)+C(cos5x )=0

Linearly independent, x = 0

Step-by-step explanation:

- We are given three functions as follows:

- We are to determine the linear - independence of the given functions. We will use the theorem of linear independence which states that:

Where,

                       c1 , c2 , c3 are all zeroes then for all values of (x),

- The system of function is said to be linearly independent

- We will express are system of equations as such:

- To express our system of linear equations we will choose three arbitrary  values of ( x ). We will choose, x = 0. then we have:

- Next choose x = 1:

- Next choose x = 2:

- Solve the last two equations simultaneously we have:

                              .... ( Only trivial solution exist )

The functions are linearly independent

- The only zero exist is x = 0.

answer: all the values are within because it has a pattern that repeats from -6 to 4.

good luck!