(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

find the local extrema of

f(x,y) = 6x^2 -8x + 2y^2 - 5 around the closed disk x^2 +y^2 =< 1

2. Relevant equations

3. The attempt at a solution

I used the larangian way of doing this but not sure if its right. My solution:

F(x,y) = f(x,y) + lambda g(x,y)

where g(x,y) = x^2 + y^2 - 1

taking partials wrt to x y and lambda I get

Fx = 12x - 8 + 2x lamda

Fy = 4y + 2y lamda

Flamda = g(x,y)

setting each to 0 and solving I get

lamda = 2 from Fy

subbing into Fx I get x = 1

subbing into g(x,y) and solving I get y = ±1

so the extrema are

(1,1) (1,-1)

edit: I think I shouldve had my equation as

F(x,y) = f(x,y) - lamda g(x,y) rather than f(x,y) + lamda g(x,y)

with that equation I get

x = 0.5

y = ±sqrt 0.75

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# Constrained extrema

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