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Find the extreme values of the function subject to the given constraint.$$f(x,\, y) = y^2 - x^2,\, x^2 + y^2 = 16$$
I understand how to to compute the extrema using Lagrange multipliers and lambda however I keep getting this question wrong. I end up with $$-2x=λ*2x$$ $$2y=λ*2y$$
1 Answer
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Your first equation $-2x=\lambda\cdot 2x$ implies that either $x=0$ or $\lambda=-1$ (or both).
Your second equation $2y=\lambda\cdot 2y$ implies that either $y=0$ or $\lambda=1$ (or both).
Combining those two, you have a total of four possibilities. Look at each possibility and find the possible points (if any) for that possibility. Then look at the function values at each resulting point. Use those to find the maximum and minimum values of your function.
