How to compute a geodesic (homework question)

Question

I am interested to compute the a geodesic in the Euclidean space $\mathbb{R}^1$. I am not familiar with differential geometry but I know that a geodesic is the solution of the following optimization problem $$ \min_{\gamma\in\Gamma(a,b)}\int_0^1\dot{\gamma}(r)^2\,dr, $$ where $\Gamma(a,b)$ is the set of piece wise continuous curves connecting $a\in\mathbb{R}$ to $b\in\mathbb{R}$.