prove the profit maximization of the consumer i.e. mu=p, accordintg to cardinalist using mathmatical derivation.

Answer 1

First we need to understand "cardinalist", someone in favour of the cardinal utility theory, namely utility (happiness) is quantifiable, [ and can be mathematically represented by a curve, and diffentiable, most of the time. ]

https://www.enotes.com/homework-help/discuss-cardinal-utility-theory-how-do-cardinalist-115449

We can consider the typical consumer to make choices in small units, and having limited budget. So each next unit of money will be spent on the most profitable (maximum utility/unit price, most happiness/unit price) item. Since utility varies (usually decreases) with quantity, the choice of the next unit of spending may be different. So the choice depends on maximum (additional) utility/unit price, or mu/p.

A more detailed explanation can be found in the following presentation (requires Microsoft Powerpoint or MPP reader).
gottheil.swcollege.com/gottheil3e/powerpoint/marg_util_powerpoint.ppt

Answer 2

We can consider the typical consumer to make choices in small units, and having limited budget. So each next unit of money will be spent on the most profitable (maximum utility/unit price, most happiness/unit price) item. Since utility varies (usually decreases) with quantity, the choice of the next unit of spending may be different. So the choice depends on maximum (additional) utility/unit price, or mu/p.