Q13/124) Let P(x) = x^3 - a x^2 + x - b Prove that there is a root between a and b (inclusive)

P(a) = a^3 - a^3 + a - b = a - b
P(b) = b^3 - a b^2 + b - b = b^2 (b - a)

If a = b then P(a) = P(b) = 0 so a and b are roots

If a and b are not same then P(a) and P(b) are of opposite signs so according to the intermediate value theorem, there exists c between a and b, such that

P(c) = 0.