Show that the least possible distance between an object and its real image in a convex lens is 4f where f is the focal length of the lens.

Suppose I is the real image of an object O. Let d be the distance between them. If the image distance is x, the object distance will be (d – x).
Thus,    u = – (d – x) and v = + x
Sustituting in the lens formula we have
                  
or,                
or,              

For a real image, the value of x must be real, i.e., the roots of the above equation must be real. This is possible if
d² ≥ 4fd
or,    d ≥ 4f
Hence, 4f is the minimum distance between the object and its real image formed by a convex lens.