When a convex lens produces a real image of an object, the minimum distance between the object and the image is equal to four times the focal length of the convex lens.

To understand this, we need to consider the behavior of light rays passing through a convex lens. When light from an object passes through a convex lens, it converges to form an image on the opposite side of the lens. The distance between the object and the lens is called the object distance, and the distance between the image and the lens is called the image distance.

In this case, when a real image is formed by a convex lens, the image is formed on the opposite side of the lens from the object. The minimum distance between the object and the image occurs when the object is placed at a distance equal to four times the focal length of the lens. This is the point where the image is the clearest and most focused.

Therefore, the correct answer is option 3 - four times the focal length of the convex lens. Remember that focal length is a property of the lens and remains constant regardless of the position of the object or the type of image formed.