<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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Answer:
7. 25 and 8. 18. I think it is. I hoped that is was right.
Step-by-step explanation:
150/6=25. 90/5=18.
Answer:
The speed of Mr Solberg's scooter when there is no wind is 35 miles per hour
Step-by-step explanation:
The given parameters are;
The time Mr. Solberg rides his scooter and cover 3 miles against the wind = The time Mr. Solberg rides his scooter and cover 4 miles with the wind
The speed of the wind = 5 miles per hour
Let v, represent the speed of Mr Solberg's scooter when there is no wind, we have;
Time = Distance/Speed
Mr Solberg's speed against the wind = v - 5
Mr Solberg's speed with the wind = v + 5
The distance covered at a given time, while riding against the wind = 3 miles
The distance covered at the same time, while riding with the wind = 4 miles
The time in both instances are therefore 3/(v - 5) = 4/(v + 5)
From which we have;
3 × (v + 5) = 4 × (v - 5)
3·v + 15 = 4·v - 20
20 + 15 = 4·v - 3·v
4·v - 3·v = 20 + 15
v = 35
The speed of Mr Solberg's scooter when there is no wind = v = 35 miles per hour.
Answer:
10.28425
Step-by-step explanation:
