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Find the greatest and smallest values that the function f(x, y) = xy takes on the ellipse x2 8 + y2 2 = 1?
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Most Upvoted Answer
Find the greatest and smallest values that the function f(x, y) = xy t...
Understanding the Problem
To find the maximum and minimum values of the function f(x, y) = xy on the ellipse defined by the equation x^2/8 + y^2/2 = 1, we can utilize the method of Lagrange multipliers or parameterize the ellipse.
Parameterization of the Ellipse
The ellipse can be parameterized as follows:
- x = sqrt(8) * cos(θ)
- y = sqrt(2) * sin(θ)
Here, θ ranges from 0 to 2π.
Substituting into the Function
Now substitute x and y into f:
- f(θ) = (sqrt(8) * cos(θ)) * (sqrt(2) * sin(θ))
- f(θ) = sqrt(16) * cos(θ) * sin(θ)
- f(θ) = 4 * cos(θ) * sin(θ)
Using the identity sin(2θ) = 2 * sin(θ) * cos(θ):
- f(θ) = 2 * sin(2θ)
Finding Maximum and Minimum Values
To find the extrema, consider the range of sin(2θ):
- The maximum value of sin(2θ) is 1.
- The minimum value of sin(2θ) is -1.
Thus:
- Maximum: f(θ) = 2 * 1 = 2
- Minimum: f(θ) = 2 * (-1) = -2
Conclusion
- Greatest Value: 2
- Smallest Value: -2
These values of f(x, y) = xy are achieved when θ = π/4 (for the maximum) and θ = 3π/4 (for the minimum).
Community Answer
Find the greatest and smallest values that the function f(x, y) = xy t...
Find the greatest and smallest values that the function
f
(
x, y
) =
xy
takes on the ellipse
x
2/
8+
y
2/
2= 1
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Which of the following statement is true for Power-of-2 allocators? Buddy System splits and recombines memory blocks in a predetermined manner during allocation and deallocation. No splitting of blocks takes place, also no effort is made to coalesce adjoining blocks to form larger blocks; when released, a block is simply returned to its free list. When a request is made for m bytes, the allocator first check the free list containing blocks whose size is 2i for the smallest value of i such that 2i ≥ m.if the free list is empty ,it checks the list containing blocks that are higher the next higher power of 2 in size on so on. an entire block is allocated to a request. When a request is made for m bytes. the system finds the smallest power of 2 that is ≥ m. Let this be 2i.if the list is empty, it checks the lists for block of size 2i+1.it takes one block off this list and splits it into two halves of size 2i.it put one of these blocks into the free list of size 2i,and uses the other block to satisfy the request.

Which of the following statement is true for buddy system allocators? Buddy System splits and recombines memory blocks in a predetermined manner during allocation and deallocation. No splitting of blocks takes place, also no effort is made to coalesce adjoining blocks to form larger blocks; when released, a block is simply returned to its free list. When a request is made for m bytes, the allocator first check the free list containing blocks whose size is 2i for the smallest value of i such that 2i ≥ m.if the free list is empty ,it checks the list containing blocks that are higher the next higher power of 2 in size on so on. an entire block is allocated to a request. When a request is made for m bytes. the system finds the smallest power of 2 that is ≥ m. Let this be 2i.if the list is empty, it checks the lists for block of size 2i+1.it takes one block off this list and splits it into two halves of size 2i.it put one of these blocks into the free list of size 2i,and uses the other block to satisfy the request.

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Find the greatest and smallest values that the function f(x, y) = xy takes on the ellipse x2 8 + y2 2 = 1?
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