Intermediate pressure in a two-stage compressor

[cshin10]

Homework Statement

The prompt says that there's a 2-stage adiabatic compression cycle where the first compressor pressurizes atmospheric air (0psig or 14.7psia) to P2, and the compressed air is cooled by a cooler to the initial temperature, and passes through another compressor to 350psig or 364.7psia. The question to calculate the optimum P2, which is the P2 that will give the lowest sum of the workload of the two compressors.

Homework Equations

|W|=αnRT₁*[(P₂/P₁)^(γ-1)/γ-1]
For ideal gas, γ=7/5, α=5/2

The Attempt at a Solution

I am supposed to derive P2 by hand and then compare it with CHEMCAD simulation results, but I am completely stuck on the hand calculation part. What I've written above is literally everything the prompt tells me... My simulation results show that a pressure ratio of 5.04 and 4.923 at each compressor gives the minimum work, which coincides with the popular heuristic for optimal P2 in a two-stage compressor. I guess ideal gas law can be used since it's just air.
I've tried doing it this way.
W₁=5/2*nRT₁*((P₂/P₁)^(2/7)-1)
W₂=5/2*nRT₁*((P₃/P₂)^(2/7)-1)
W_Total=W₁+W₂
=5/2*nRT₁[(P₂/P₁)^(2/7)+(P₃/P₂)^(2/7)-2)

So the minimum of (P2/14.7)^(2/7)+(364.7/P2)^(2/7)-2 would give the optimum P2 value but this thing doesn't have an absolute minimum and I am stuck...

 

[mfb]

It might not have an absolute minimum (did not check this) but it has exactly one relative minimum in the relevant pressure range.