Consider a multi$-$stage compression system containing three compressors and two intercoolers. The ideal work of each compression cycle $($see Chapter $4),$ gives:

$W_c=\frac{kR{T}_i}{k-1}[(\frac{p_o}{p_i}{)}^{\frac{k-1}{k}}-1]$

If $p_1=100$kPa and $p_4=1000$kPa, what are the optimal values of $p_2$ and $p_3$ such that the total work required is a minimum. Assume $k=1\mathrm{.}4$ for air and $T_1=298K.$ Verify the solution is indeed a minimum.