Convex Mirrors magnification help

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The magnification of a convex mirror is always positive and greater than zero, which is determined by the negative focal length. The size of the magnification does depend on the object distance, as demonstrated through ray diagrams showing different image sizes for the same object at varying distances. The relationship for magnification can be derived from the formula 1/u + 1/v = 1/f, with magnification defined as M = -v/u. It is confirmed that the magnification for a convex mirror cannot equal 1; it can only be greater than 1. Understanding these principles is essential for analyzing convex mirrors effectively.
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I know that magnification of a single convex mirror is always positive (that is, greater than 0) based on f<0, but does the size of the magnification depend on the magnitude of the object distance?
 
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The simplest way to answer this question (yourself) is to draw the ray diagrams for a convex mirror putting the same (sized) object at two different positions, and see if they lead to different image sizes.

The other way to answer this question is to derive the relation for the magnification from the formula :
1/u + 1/v = 1/f. This isn't terribly hard, if you remember that the magnification, M = -v/u
 
Can m for a convex mirror ever be equal to 1 aside from being greater than 1?
 
Can m for a convex mirror ever be equal to 1 aside from being greater than 1?

nope, as you said in the original post
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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