Image can be formed on the screen if it is real. Real image of reduced size can be formed by a concave mirror or a convex lens

Let$u=2f+x$, then

$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$

$\Rightarrow \frac{1}{2f+x}+\frac{1}{v}=\frac{1}{f}$

$\Rightarrow \frac{1}{v}=\frac{1}{f}-\frac{1}{2f+x}=\frac{f+x}{f(2f+x)}$

$\Rightarrow v=\frac{f(2f+x)}{f+x}$

It is given that $u+v=1.0 \mathrm{m}$

$2f+x+\frac{f(2f+x)}{f+x}=\left(2f+x\right)\left[1+\frac{f}{f+x}\right]<1.0$m

Or $\frac{{\left(2f+x\right)}^2}{f+x}<1.0$m

Or ${ \left(2f+x\right)}^2<(f+x)$

This will be true only when $f<0.25 \mathrm{m}$