This is an interesting problem of a newsboy, the problem is to identify the optimal quantity he has buy one fine day, the below equation is used for calculating the optimal quantity Q?
the below equation is used to calculate the optimal quantity Q
$$(Eq. 1)->P(Q)=(p_s - p_r + p_g)\int_{0}^{Q}{xf(x)} dx + p_rQF(Q) - p_oQ + (p_g + p_s)Q(1-F(Q)) - p_g\bar{D}$$ , where
$$p_s -> purchase price$$
$$p_r -> salvage price$$
$$pg -> goodwill price$$
and Demand $$ \bar{D} = \int_{-\infty}^{\infty}{xf(x)dx}$$
The above (Eq. 1) is concave in Q for any density function f(x).
To compute the optimal quantity Q*, it said that by invoking Fermat’s theorem, and setting $$dP(Q)/dQ = 0$$, we get an distribution independent result (independent of f(x))
$$Q^* = F^{-1} \left(\frac{p_s+p_g-p_o} {p_s+p_g-p_r}\right)$$
Can some one help me to understand what Fermat’s theorem is about and when is it used, also how is it used to simplify (Eq. 1)
Thanks, Kamal.