Find the maximum value, M, of the function f(x,y) = x^4 y^9 (7 - x - y)^4 on the region x = 0, y = 0, x + y

Question

Find the maximum value, M, of the function f(x,y) = x^4 y^9 (7 - x - y)^4 on the region x >= 0, y >= 0, x + y <= 7

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s3a · Answer

Firstly, my handwriting is ugly because I wrote this for myself before thinking that I would need to post it online but I think it should be legible nonetheless. If it isn't, tell me and I will rewrite it from scratch.

Secondly, the question doesn't force me to use the Lagrange Multiplier method but I chose it thinking it's the best way since it seems like a nice method so if I am wrong in choosing it, tell me. Using the Lagrange Multiplier method, I get M = 0. That's an extremum alright but it's a minimum and not a maximum given the set of constraints.

Any input in helping me figure out what I did wrong would be greatly appreciated!
Thanks in advance!

anonymous · Answer

have you considered using partial derivatives then it should be able to identify the max./mins fo the equation