Let's break this down step by step as requested: --- ### **Step 1: Set up the linear combination** Given functions: - \( f_1(x) = x \) - \( f_2(x) = x^2 \) - \( f_3(x) = x^3 \) We want to find constants \( c_1, c_2, c_3 \) (not all zero) such that: \[ g(x) = c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \quad \forall x \in (-\infty, \infty) \] Substitute the given functions: \[ g(x) = c_1 x + c_2 x^2 + c_3 x^3 = 0 \quad \forall x \] **Explanation:** This equation must hold for all \( x \). This means the coefficients of every power of \( x \) must be zero. --- ### **Step 2: Set coefficients to zero and solve for \( c_1, c_2, c_3 \)** \[ c_1 x + c_2 x^2 + c_3 x^3 = 0 \implies \begin{cases} c_1 = 0 \\ c_2 = 0 \\ c_3 = 0 \\ \end{cases} \] **Explanation:** A polynomial is identically zero for all \( x \) only if all its coefficients are zero. Thus, the only solution is \( c_1 = 0, c_2 = 0, c_3 = 0 \). --- ### **Final Answers:** **1. Solution for \( (c_1, c_2, c_3) \):** \[ (0, 0, 0) \] **2. Determine if the functions are linearly independent:** The only solution is the trivial solution, so the functions are **linearly independent** on \( (-\infty, \infty) \). **Boxed answers:** - \( (c_1, c_2, c_3) = (0, 0, 0) \) - Linearly independent (select this option)