Linear Dependence of Functions with Absolute Value

[jinksys]

Homework Statement

Determine whether the pairs of functions are linear dependent or linearly independent.

f(x) = x^3
g(x) = x2|x|

Homework Equations

The Attempt at a Solution

g(x)=x^2|x| = x^2*sqrt[x^2] = sqrt[x^6] = x^3

f'=3x^2
g'=3x^2

fg'-f'g = 0

Linearly Dependent according to me, Linearly independent according to the book.

I assume it has to do with the absolute value, could someone enlighten me?

 

[jinksys]

Should g'(x) be 2x|x|+x^2(|x|/x) ?

 

[rsa58]

note that f is positive for some values of x and negative for others whereas g is always positive. if these two were linearly dependent one would be a constant multiple of the other FOR ALL VALUES OF x. can we multiply a fully positive function by a number so that part of it becomes negative? no. therefore no linear dependence. note also this depends on the set over which g and f are defined.

 

[rsa58]

why are you take the derivative?