Plan:
- Differentiate p(s)
- Solve for 0
- Compare min to p(50) & p(75)

p[s_] := 10/Sqrt[s] + (s/100)^3

p[50]

N[(1/8)+(Sqrt[2])]

p[75]

N[(27/64)+(2/(Sqrt[3]))]

Solve[ p'[s]==0, s]

N[{{s->((50 (2^(5/7)))/(3^(2/7)))},{s->-((50 (((-1))^(3/7)) (2^(5/7)))/(3^(2/7)))},{s->((50 (((-1))^(4/7)) (2^(5/7)))/(3^(2/7)))}}]

p[59.9337]