Final answer:
To check if the set {3x^2+4x-2, 2x^2-x+3, 7x^2+2x+4} is linearly independent, we use the Wronskian determinant. If the determinant is nonzero, the set is independent. In this case, we calculate the Wronskian directly from constructing a 3x3 matrix of the functions' coefficients.
Step-by-step explanation:
The Wronskian is a determinant used to determine the linear independence of a set of functions. For a set of functions {f(x), g(x), h(x)}, if the Wronskian is non-zero for all values of x in the given interval, then the set is linearly independent.
For the set {3x^2+4x-2, 2x^2-x+3, 7x^2+2x+4}, let the functions be denoted as f(x), g(x), and h(x), respectively.
The Wronskian (W) is calculated as follows:
W = f & g & h \\ f' & g' & h' \\ f'' & g'' & h''
Here, f', g', and h' are the first derivatives, and f'', g'', and h'' are the second derivatives.
W = 3x^2+4x-2 & 2x^2-x+3 & 7x^2+2x+4 \\ 6x+4 & 4x-1 & 14x+2 \\ 6 & 4 & 14
Now, calculate the determinant:
W = (3x^2+4x-2)((4x-1)(14) - (14x+2)(4)) - (2x^2-x+3)((6x+4)(14) - (14x+2)(6)) + (7x^2+2x+4)((6x+4)(4) - (4x-1)(6))
Simplify the expression and evaluate for x:
W = (3x^2+4x-2)(56 - 56x - 8) - (2x^2-x+3)(84 - 84x - 12) + (7x^2+2x+4)(24 - 24x + 6)
W = (3x^2+4x-2)(-8 - 56x) - (2x^2-x+3)(-12 - 84x) + (7x^2+2x+4)(30 - 24x)
W = (-24x^3 - 44x^2 - 16x + 16) - (-24x^3 - 8x^2 + 168x + 36) + (210x^2 + 92x - 120)
Combine like terms:
W = 186x^2 + 140x - 140
The Wronskian is a non-zero constant, indicating that the set {3x^2+4x-2, 2x^2-x+3, 7x^2+2x+4} is linearly independent on (-∞, ∞).