# Analytic maximization that reduce fails to solve

I'm working on an interesting problem but stuck on a step that Mathematica fails on (no matter what angle I try to get at it from).

I'm attempting to find the symbolic maximum of this convex function (with respect to DY):

PArbNegDY[K0_, Y0_, S0_, DY_, DS_, \[Lambda]_] := -DY - (
K0 (DS + S0) (-1 + (Y0/(DY + Y0))^(1/\[Lambda])))/Y0

K0>0 && Y0>0 && S0>0 && DS>-S0 && 0<\[Lambda]<=1


So, I differentiated with respect to DY and am now trying to solve for it:

0 == -1 + (K0 (DS + S0) (Y0/(DY + Y0))^(-1 + 1/\[Lambda]))/((DY + Y0)^2 \[Lambda])


However, neither solve nor reduce give an answer (reduce runs forever). Any help would be greatly appreciated.

• There might not be a reasonably simple solution when the values of the other parameters are unknown. Plugging in values you used in the figure for K0, DS, S0, Y0, and $\lambda$ results in a solution. – JimB Aug 16 at 22:26
• Do you have any reason to think there is a closed form for transcendental equation?. Mathematica fails,because mathematics FAILS.Try: Maximize[{PArbNegDY[K0, Y0, S0, DY, DS, 2/3], K0 > 0 && Y0 > 0 && S0 > 0 && DS > -S0}, DY] // MatrixForm – Mariusz Iwaniuk Aug 17 at 10:35