Note you have a small mistake

your term P(L,2L) = 100L^0.25 x 2L^0.75 = 10 000

should be P(L,2L) = 100L^0.25 x (2L)^0.75 = 10 000

2^(.75)L =100

L = 100/2^(.75) = 59.5

K = 119

This has to be a minimum since there is no max

Consider

C(L,K) = 40L + 60K and P(L,K) = 100L^0.25 x K^0.75 = 10 000

as L goes to 0 k goes to infinity and C goes to infinity

similarly if k goes to 0

Also to convince yourself solve for k in terms of L and plug this into C and graph--again you'll see this is a minimum