Compute robust linear interpolation with edge cases

You are given two distinct points (x0, y0) and (x1, y1) with x0 != x1. a) Derive the linear interpolation function f(x) on [min(x0, x1), max(x0, x1)] and compute f(19) when (x0, y0) = (10, 100) and (x1, y1) = (25, 175). Provide both the exact rational form and a decimal rounded to 3 decimals. b) Prove that f(x) can be written as a convex combination of y0 and y1 via t = (x − x0)/(x1 − x0). c) Specify a consistent rule for the degenerate case x0 = x1 with y0 != y1 (e.g., error, average, or limiting value) and justify your choice. d) If the observed y-values have independent bounded errors |ε| ≤ 0.5, give a bound on the interpolation error at any x. e) Using piecewise-linear interpolation on the sorted points (0,0), (3,9), (8,4), (10,10), compute f(6) and f(9), and state whether each evaluation is interpolation or extrapolation.