<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>
</span>
Answer:
x = 20
Step-by-step explanation:
The marked angles are vertical angles and congruent, thus
6x - 10 = 3x + 50 ( subtract 3x from both sides )
3x - 10 = 50 ( add 10 to both sides )
3x = 60 ( divide both sides by 3 )
x = 20
Answer:
B
M=2
Step-by-step explanation:
rise over run
you rise 2 and go over 1
2 over 1 is just 2.
Answer:
r<4
Step-by-step explanation:
-3(r-4)>0 step 1: distribute the -3
-3r+12>0 explain: -3 times r =-3r, -3 times -4 = 12
next, isolate the -3r by putting 12 on the other side
-3r+12>0 subtract 12 from both sides
-3r>-12 now, divide both sides by -3 (you have to flip the > because you are dividing by a negative) -12 divided by -3 is 4
so you should end up with r<4
B is the answer b.converse
