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max value of f=x+y+z

on the ball x^2+y^2+z^2=a^2 (not inside the ball, only on it)

a>0

with contidions

x<=0

y<=0

z<=0

is equal to?

i get -sqrt(3)a

the right answer is -a

help me plz

i'll say what ive done:

grad(x+y+z)+j*grad(x^2+y^2+z^2-a^2)=(0,0,0)

x^2+y^2+z^2=a^2

(j symbolies gama)

for example from the x equation we get:

1+2jx=0

x=-1/(2j)

same with y&z

when we put those x,y,z at the ball equation we get

j^2=3/(4a^2)

j=sqrt(3)/(2a)

therefore:

x=y=z=-1/(2j)=-a/sqrt(3)

therefore

max value: -sqrt(3)*a

where's my mistake?

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# Lagaranze multiplayers

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