Final answer:
The functions f(x) = x² and g(x) = x³ are shown to be linearly independent by computing their Wronskian, which does not vanish identically and therefore confirms their linear independence.
Step-by-step explanation:
To show that the functions f(x) = x² and g(x) = x³ are linearly independent using the Wronskian, we first need to compute the Wronskian determinant of f and g. The Wronskian W(f, g) is given by:
W(f, g) = | f(x) f'(x) |
| g(x) g'(x) |
Plugging the functions into the determinant, we get:
W(f, g) = | x² 2x |
| x³ 3x² |
Calculating the determinant, we have:
W(f, g) = x²(3x²) - x(2x) = 3x⁴ - 2x²
Simplifying this, we obtain:
W(f, g) = x²(3x² - 2)
Since the Wronskian is a function that does not vanish identically (it is not always zero for all x), we can conclude that f(x) = x² and g(x) = x³ are linearly independent functions.