IVT c squared Question

Question

Suppose $g:R\to R$ is continuous on $R$ s.t $g(1)=4$ and $g(2)=1$. Does there exist a $c\in R$ such that $g(c)=c^2$

Proof

Yes, there is. let $f(x)=x^2-g(x)$ First note $f$ is a continuous function, difference of 2 functions $x^2$ and $g(x)$