Let P3(x) be the interpolating Lagrange polynomial of degree 3 passing through the points (0,0), (1/2, y), (1,3) and (2,2). If the coefficient of x3 in P3(x) is 6, then determine the value of y.

Assume the polynomial P5(x) interpolates a function f(x) at the six data points (xi, f(xi)) with x-coordinates ₁ = 0.0, 2 0.0, x2 = 0.2, x3 = 0.4, x4 = 0.6, x5 = 0.8, x6 = 1.0. Assume that the interpolation error at x = 0.3 is |f(0.3) - P5(0.3)| = 0.01. Estimate the new interpolation error |ƒ(0.3) — P7(0.3)| that would result if two additional interpolation points - (x7, y7) = (0.1, f(0.1)) and (x8, ys) = (0.5, f(0.5)) are added. What assumptions have you made to produce this estimate?

4. Prove that f(x) = 4x³ – 8x2 + 2x + 1 has three real zeros. %3D