Question

Let \( \alpha, \beta, \gamma, \delta \in \mathbb{R} \) be such that the quadrature formula \[ \int_{-1}^{1} f(x) \, dx = \alpha f(-1) + \beta f(1) + \gamma f'(-1) + \delta f'(1) \] is exact for all polynomials of degree less than or equal to 3. Then, \( 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \) is equal to (in integer):

Hint:

Use the exactness condition of the quadrature formula for polynomials up to degree 3 to form equations for the weights \( \alpha, \beta, \gamma, \delta \). Solve the system to find the final result.

Answer 1

Step 1: Understanding the Problem
The given quadrature formula involves the values of the function \( f \) and its derivative \( f' \) at \( x = -1 \) and \( x = 1 \). To ensure the formula is exact for all polynomials up to degree 3, we use polynomials of degree 1, 2, and 3 to derive relationships for \( \alpha, \beta, \gamma, \delta \). 
Step 2: Applying Exactness Conditions
For exactness, the quadrature formula must exactly integrate polynomials of degree up to 3. We will apply this condition using specific test functions (polynomials) and then equate the results. For polynomials of degree 1, 2, and 3, we calculate integrals on the left-hand side and equate them with the quadrature formula on the right-hand side. This gives us a system of equations involving \( \alpha, \beta, \gamma, \delta \). 
Step 3: Solving the System of Equations
By solving the system of equations derived from the exactness conditions, we obtain the values of \( \alpha, \beta, \gamma, \delta \). 
Step 4: Final Computation Once we have the values of \( \alpha, \beta, \gamma, \delta \), we calculate \( 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \). \[ 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) = 20 \] Thus, the value is \( \boxed{20} \). % Final Answer \[ \boxed{20} \]