Hello! I have started on the problem below but am now completely stuck! Can someone please help me out? This is revolving around The Producer's Cost Minimization Problem in microeconomics. I have outlined my current answers below in bold if you can hep me get on the right path.

Assumption: We have a firm with the production function f(z1,z2)=z1*z2 as well as per unit prices w1 and w2 for their inputs:

a) What is the firm's marginal rate of technical substitution at input bundle (z1,z2)? I have z2/z1 as the marginal rate of technical substitution

b) What are all the bundles (z1,z2) with MRTS(z1,z2)=1? I wasn't sure on this one, but I believe it is z1, z2>0 as well as z1=z2?

c) What are all the bundles (z1,z2) with MRTS(z1,z2)=4? I also wasn't sure here as well, I'm thinking it could be z1=4*z2 where z1 & z2>0

d) With w1=w2=1 suppose the firm's target output varies. What is the firm's long-run expansion path (i.e. the path of input choices as target output varies)? I'd assume this is best represented on a graph where we start at the origin and movement along the line gives us the costs at which output can be expanded but I have no idea what to put here.

e) Explain why there is no additional work required to determine the firm's cost minimizing input levels z*1(y;2,2) and z*2(y;2,2) when prices double from i.e. w1=w2=1 to w1=w2=2. Honestly no idea here.