Given f(x) = 3 - 5x + x^3, how can I show that f(x) = 0 has a root (x=a) in the interval 1<a<2?
In plain english, we need to show that there is a value of x, which we call "a", in the interval 1 < a < 2 where f(a)=0. To prove this we start by letting x = 1: f(1) = 3 - 5(1) + 13 = -1. We now let x = 2: f(2) = 3 - 5(2) + 23 = 1. Since there is a change of sign of the value of f(x) in the interval of 1 < x < 2, then there must be a value of x = a where f(a) is zero. Therefore, the function f(x) = 0 has a root (x = a) in the interval 1 < a < 2.
