Show that the equation x3 14x c 0 has at most one solution

Last updated: 7/2/2023

Show that the equation x3 14x c 0 has at most one solution in the interval 2 2 Let f x x 14x c for x in 2 2 If f has two real solutions a and b in 2 2 with a b then Select implies that there is a number r in a b such that f r 0 Select Now f r equation cannot have two real solutions in 2 2 Hence it has at most one real solution in 2 2 Since the polynomial f is continuous on a b and differentiable on a b Since r is in a b which is contained in 2 2 we have r 2 so r2 4 It follows that 3r2 1473 4 14 2 0 This contradicts f r 0 so the gi