<span>We want to optimize f(x,y,z)=x^2 y^2 z^2, subject to g(x,y,z) = x^2 + y^2 + z^2 = 289.
Then, ∇f = λ∇g ==> <2xy^2 z^2, 2x^2 yz^2, 2x^2 y^2 z> = λ<2x, 2y, 2z>.
Equating like entries:
xy^2 z^2 = λx
x^2 yz^2 = λy
x^2 y^2 z = λz.
Hence, x^2 y^2 z^2 = λx^2 = λy^2 = λz^2.
(i) If λ = 0, then at least one of x, y, z is 0, and thus f(x,y,z) = 0 <---Minimum
(Note that there are infinitely many such points.)
(f being a perfect square implies that this has to be the minimum.)
(ii) Otherwise, we have x^2 = y^2 = z^2.
Substituting this into g yields 3x^2 = 289 ==> x = ±17/√3.
This yields eight critical points (all signage possibilities)
(x, y, z) = (±17/√3, ±17/√3, ±17/√3), and
f(±17/√3, ±17/√3, ±17/√3) = (289/3)^3 <----Maximum
I hope this helps! </span><span>
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I believe you are asking in how many ways they can sit. If so:
The 1st can sit anywhere: he has only 1 way to sit
The 2nd can sit in 11 ways, since one seat is already occupied
The 3rd can sit in 10 ways, since 2 seat are already occupied
The 4th can sit in 9 ways, since 3 seat are already occupied
The 5th can sit in 8 ways, since 4 seat are already occupied
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The 12th can sit in 1 way, since11 seat are already occupied
General formula for a circular table:
Number of ways they n persons can be seated: (n-1)!
and the 12 can be seated in (12-1)! = 11! = 39,916,800 ways.
This is called circular permutation
Answer: 1.5 times
3 flips divided by 2 sides of a coin would be 1.5
this could be wrong
Step-by-step explanation:
a) f(1) = -1 → x = -1
b) g(1) = Undefined
c) f(x) = 1 → x= Undefined
d) g(x) = 1 → x= 5
Answer:
Step-by-step explanation:
They are about 3 meters away from each other
and the correct area is 15.5-12 and the correct answer is 3.5 meters apart
