Large imperfect fields are existentially closed in function fields after finite constant extension

For an algebraic function field $F$ over a large field $K$, we show: 1) if $F\mathrm{\mid }K$ has a rational place, then there is a finite purely inseparable extension $K^{\mathrm{\prime }}\mathrm{\mid }K$ such that $K^{\mathrm{\prime }}$ is existentially closed in $F\mathrm{.}{K}^{\mathrm{\prime }}$; 2) $F\mathrm{\mid }K$ has a rational place admitting local uniformization if and only if $K$ is existentially closed in $F$.