The equation " x^3-3x+1=0 " has three real roots. Show that one of the roots lies between −2 and −1

A simple way to prove this is to sub in the values that we are given. so f(x) will represent our equation x^3-3x+1 (that is f(x) = x^3-3x+1)f(-2) = -1 < 0f(-1) = 3 > 0The first thing we notice is that both answers are either side of zero. this is good as it indicates that if we where to graph the curve then one point will be at exactly zero and hence a root. For our previous statement to be correct we just need to know that the curve is continuous which it is. so hence this proves that there is a root between our two values