While reading Intermediate Microeconomics from Hal Varian, I fell short in understanding the relationship between SRMC and LRMC. I can see how in SR, when fixed factor is chosen at LR optimizing condition, then
c(y) = cs (y,k(y))
where, k is the fixed factor.
But at pg. 394 of 8th edition, in Appendix to Ch. 21 (Cost Curves), author writes, "...the long-run marginal cost will consist of two pieces: how costs change holding plant size fixed plus how costs change when plant size adjusts. But if the plant size is chosen optimally, this last term has to be zero!". This is what I am unable to understand, how the 'second' term is zero!
Please, help me understand both intuitively and through calculus.