r/HomeworkHelp • u/No-Beginning-8506 Secondary School Student • 6d ago
Additional Mathematics [Multivariate Calculus: Local Extrema]
I’ve been working on this one question and can’t seem to get past step 1. Either I derived my equations wrong or I just don’t know how to find the zeros, or both. I keep getting to this weird quadratic formula thing with y as a constant and vice versa, not really knowing where to go from there. I also started finding the true derivative but sorta gave up on it since I didn’t have my critical points anyway…
I’d appreciate if someone walked me through finding the critical points, since even if I found the solutions to each partial derivative I’m not quite sure how to turn that into a coordinate. Additionally, I’m not super confident on the first part of the question, using level curves, which is why I skipped it. How would you make sketches of this graph? I feel like it’s too ugly for me to comprehensively turn into something I can work with. Anything is appreciated!
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u/Alkalannar 6d ago
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Find df/dx. For df/dx = 0, you get y = x - 1/2x.
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Find df/dy. For df/dy = 0, you get x = y - 1/2y.
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Combine together: x = (x - 1/2x) - 1/2(x - 1/2x).
This actually becomes a quadratic that you can solve. You'll get x = a and x = b as solutions. -
But y is exactly the same! You get y = a and y = b as solutions.
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So you need to check (a, a), (a, b), (b, a), and (b, b) as your critical points.
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For a max, you need both d(df/dx)/dx and d(df/dy)/dy to be negative. For a min, you need them both positive. For a saddle point, you need one of each.
So what are d(df/dx)/dx and d(df/dy)/dy?
What are df/dx and df/dy to begin with?
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u/No-Beginning-8506 6d ago
Thank you! You helped me realize I was dropping some terms and doing some algebra wrong, I got it finally!
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u/fletchman93 6d ago
Before you factored, the last term in your partial w.r.t. y has the wrong sign.
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