Question

Show that the equation x^3+2x=x^2+1 has at least one solution on (0,1). I get 0=1 proving that it doesn't have a solution but I want to be sure there isn't something else to it.

Answer 1

Let \(f(x)=x^3+2x\), and \(g(x)=x^2+1\). You want to show there exists \(x_0 \in (0,1)\) such that $$f(x)=g(x)\iff f(x)-g(x)=0.$$ Since f and g are continuous, we can use the intermidiate value theorem. This the same as showing that: $$f(a)-g(a)<0 \quad\text{and}\quad f(b)-g(b)>0 $$ for some \(a,b\in (0,1)\).