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Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.?
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Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0...
Maximum and Minimum Values of f on each Edge of the Square:
To find the maximum and minimum values of the function f(x, y) on each edge of the square, we need to substitute the values of x and y that lie on each edge into the function and evaluate the function.
Edge 1: x = 0, 0 ≤ y ≤ 1
Substituting x = 0 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(0, y) = -3y^2
The maximum value of f(0, y) occurs when y = 0, which gives f(0, 0) = 0.
The minimum value of f(0, y) occurs when y = 1, which gives f(0, 1) = -3.
Edge 2: x = 1, 0 ≤ y ≤ 1
Substituting x = 1 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(1, y) = 4 - 3y^2 + 2y
To find the maximum and minimum values of f(1, y), we can take the derivative of f(1, y) with respect to y and set it equal to zero:
df(1, y)/dy = -6y + 2 = 0
Solving for y, we get y = 1/3.
Substituting y = 1/3 into f(1, y), we get:
f(1, 1/3) = 4 - 3(1/3)^2 + 2(1/3) = 4 - 1 + 2/3 = 11/3
The maximum value of f(1, y) occurs when y = 1/3, which gives f(1, 1/3) = 11/3.
The minimum value of f(1, y) occurs when y = 0 or y = 1, which gives f(1, 0) = 4 and f(1, 1) = 3.
Edge 3: y = 0, 0 ≤ x ≤ 1
Substituting y = 0 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(x, 0) = 4x^2
The maximum value of f(x, 0) occurs when x = 1, which gives f(1, 0) = 4.
The minimum value of f(x, 0) occurs when x = 0, which gives f(0, 0) = 0.
Edge 4: y = 1, 0 ≤ x ≤ 1
Substituting y = 1 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(x, 1) = 4x^2 - 3 + 2x
To find the maximum and minimum values of f(x,
To find the maximum and minimum values of the function f(x, y) on each edge of the square, we need to substitute the values of x and y that lie on each edge into the function and evaluate the function.
Edge 1: x = 0, 0 ≤ y ≤ 1
Substituting x = 0 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(0, y) = -3y^2
The maximum value of f(0, y) occurs when y = 0, which gives f(0, 0) = 0.
The minimum value of f(0, y) occurs when y = 1, which gives f(0, 1) = -3.
Edge 2: x = 1, 0 ≤ y ≤ 1
Substituting x = 1 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(1, y) = 4 - 3y^2 + 2y
To find the maximum and minimum values of f(1, y), we can take the derivative of f(1, y) with respect to y and set it equal to zero:
df(1, y)/dy = -6y + 2 = 0
Solving for y, we get y = 1/3.
Substituting y = 1/3 into f(1, y), we get:
f(1, 1/3) = 4 - 3(1/3)^2 + 2(1/3) = 4 - 1 + 2/3 = 11/3
The maximum value of f(1, y) occurs when y = 1/3, which gives f(1, 1/3) = 11/3.
The minimum value of f(1, y) occurs when y = 0 or y = 1, which gives f(1, 0) = 4 and f(1, 1) = 3.
Edge 3: y = 0, 0 ≤ x ≤ 1
Substituting y = 0 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(x, 0) = 4x^2
The maximum value of f(x, 0) occurs when x = 1, which gives f(1, 0) = 4.
The minimum value of f(x, 0) occurs when x = 0, which gives f(0, 0) = 0.
Edge 4: y = 1, 0 ≤ x ≤ 1
Substituting y = 1 into the function f(x, y) = 4x^2 - 3y^2 + 2xy, we get:
f(x, 1) = 4x^2 - 3 + 2x
To find the maximum and minimum values of f(x,
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Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.?
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Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.? for Electronics and Communication Engineering (ECE) 2025 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.?.
Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.? for Electronics and Communication Engineering (ECE) 2025 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f(x, y) = 4x2 搰 3y2 2xy over the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the maximum and minimum values of f on each edge of the square. (b) Find the maximum and minimum values of f on each diagonal of the square. (c) Find the maximum and minimum values of f on the entire square.?.
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