A body of mass $m$ is kept on a rough horizontal surface (coefficient of friction = $\mu$). A horizontal force is applied to the body, but it does not move. The resultant of normal reaction and the frictional force acting on the object is given by $\vec{F}$ where:

Normal reaction $N=mg$.Frictional force $f$ opposes the applied force. Since the body doesn't move, $f$ equals the applied force, and $f\le f_{max}$ where $f_{max}=\mu N=\mu mg$.The resultant force $\vec{F}$ is the vector sum of $\vec{N}$ and $\vec{f}$.Since they are perpendicular: $|\vec{F}|=\sqrt{N^2+f^2}$.As $f_{min}=0$, $F_{min}=\sqrt{(mg{)}^2+0}=mg$.As $f_{max}=\mu mg$, $F_{max}=\sqrt{(mg{)}^2+(\mu mg{)}^2}=mg\sqrt{1+{\mu }^2}$.So, $mg\le |\vec{F}|\le mg\sqrt{1+{\mu }^2}$.The standard option provided in such constraints is the upper bound.