Maximal ideals of $F[x]/(x^2)$

Question

I'm trying to find all the maximal ideals of $F[x]/(x^2)$, where F is a field. I know that these are in bijective correspondence with the maximal ideals of $F[x]$ containing $(x^2)$, and the only maximal ideal of $F[x]$ containing $(x^2)$ is $(x)$ (since $F[x]$ is a PID and so all the maximal ideals all generated by irreducible monic polynomials in $F[x]$). Now I'm not able to understand how to take this ideal to an ideal in $F[x]/(x^2)$. More specifically, will the only maximal ideal in $F[x]/(x^2)$ be the image of $(x)$ under the canonical map $\phi: F[x] \rightarrow F[x]/(x^2)$? If yes, how do I find this image?