Establish the formula in the following two ways, using the unevenly spaced poin...

Establish the formula

in the following two ways, using the unevenly spaced points x₀ < x₁ < x₂, where x₁ − x₀ = h and x₂ − x₁ = αh. Notice that this formula reduces to the standard central-difference formula (20) when α = 1.

a. Approximate f (x) by the Newton form of the interpolating polynomial of degree 2.

b. Calculate the undetermined coefficients A, B, and C in the expression f ′′(x) ≈ Af (x₀)+B f (x₁)+C f (x₂) by making it exact for the three polynomials 1, x−x₁, and (x − x₁)² and thus exact for all polynomials of degree?2.