Question

Find a polynomial p(x) of degree $\leq$ 2 that satisfies $p(x_0) = a$, $p(x_1) = b$, and $p'(x_1) = c$ where a, b, c are given constants and $x_0, x_1$ are two different points.

Thank you!

Answer 1

p(x) = a_0 + a_1 x + a_2 x^2 p(x_0) = a_0 + a_1 x_0 + a_2 x_0^2 = a p(x_1) = a_0 + a_1 x_1 + a_2 x_1^2 = b p'(x) = a_1 + 2a_2 x p'(x_1) = a_1 + 2a_2 x_1 = c a_1 = c - 2a_2 x_1a_0 + (c - 2a_2 x_1)x_0 + a_2 x_0^2 = a a_0 + x_1 + a_2 x^2_1 = b a_0 + (x_0 + (x_0^2 - 2x_1 x_0)a_2 = a rightarrow (2) a_0 + (x_1 + (x_1^2 - 2x_1^2) a_2 = b a_0 + (x_1 - x_1^2 a_2 = b rightarrow (3) (2) - (3) c(x_0 - x_1) + (x_0^2 - 2x_1 x_0 + x_1^2) a_2 = a - b Rightarrow c(x_0 - b)+ (x_0 - b)^2 = a - b Rightarrow a_2 = (a - b) - (x_0 - x_1)/(x_0 - x_4)^2 a_0 = b + x_1^2 a_2 - cx_1 a_1 = c - 2a_2 x_1