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Consider the function f(x, y) = 4x 2 − 3y 2 + 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) = 4x 2 − 3y 2 + 2xy over the unit square...
Maximum and Minimum Values of f on Each Edge of the Square:
To find the maximum and minimum values of f on each edge of the square, we need to substitute the values of x and y for each edge and evaluate the function f(x, y).
1. Edge 1: x = 0 and 0 ≤ y ≤ 1
- Substitute x = 0 into the function: f(0, y) = 4(0)² - 3y² + 0 = -3y²
- The maximum value occurs when y = 0, which gives f(0, 0) = 0.
- The minimum value occurs when y = 1, which gives f(0, 1) = -3(1)² = -3.
2. Edge 2: x = 1 and 0 ≤ y ≤ 1
- Substitute x = 1 into the function: f(1, y) = 4(1)² - 3y² + 2y = 4 - 3y² + 2y
- To find the maximum and minimum values, we need to find the critical points by taking the derivative with respect to y and setting it equal to zero.
- f'(1, y) = -6y + 2 = 0
- Solving for y, we get y = 1/3.
- Substitute y = 1/3 into the function: f(1, 1/3) = 4 - 3(1/3)² + 2(1/3) = 4 - 1 + 2/3 = 5/3.
- The maximum value occurs at (1, 1/3) with f(1, 1/3) = 5/3.
- To find the minimum value, we evaluate the function at the endpoints of the edge:
- f(1, 0) = 4(1)² - 3(0)² + 2(0) = 4.
- f(1, 1) = 4(1)² - 3(1)² + 2(1) = 0.
- The minimum value occurs at (1, 1) with f(1, 1) = 0.
3. Edge 3: y = 0 and 0 ≤ x ≤ 1
- Substitute y = 0 into the function: f(x, 0) = 4x² - 3(0)² + 2(0) = 4x².
- To find the maximum and minimum values, we need to find the critical points by taking the derivative with respect to x and setting it equal to zero.
- f'(x, 0) = 8x = 0
- Solving for x, we get x = 0.
- Substitute x = 0 into the function: f(0, 0) = 4(0)² - 3(0)² = 0.
- The maximum and minimum values both occur at (0, 0) with f(0, 0) = 0.
4. Edge 4: y = 1 and
To find the maximum and minimum values of f on each edge of the square, we need to substitute the values of x and y for each edge and evaluate the function f(x, y).
1. Edge 1: x = 0 and 0 ≤ y ≤ 1
- Substitute x = 0 into the function: f(0, y) = 4(0)² - 3y² + 0 = -3y²
- The maximum value occurs when y = 0, which gives f(0, 0) = 0.
- The minimum value occurs when y = 1, which gives f(0, 1) = -3(1)² = -3.
2. Edge 2: x = 1 and 0 ≤ y ≤ 1
- Substitute x = 1 into the function: f(1, y) = 4(1)² - 3y² + 2y = 4 - 3y² + 2y
- To find the maximum and minimum values, we need to find the critical points by taking the derivative with respect to y and setting it equal to zero.
- f'(1, y) = -6y + 2 = 0
- Solving for y, we get y = 1/3.
- Substitute y = 1/3 into the function: f(1, 1/3) = 4 - 3(1/3)² + 2(1/3) = 4 - 1 + 2/3 = 5/3.
- The maximum value occurs at (1, 1/3) with f(1, 1/3) = 5/3.
- To find the minimum value, we evaluate the function at the endpoints of the edge:
- f(1, 0) = 4(1)² - 3(0)² + 2(0) = 4.
- f(1, 1) = 4(1)² - 3(1)² + 2(1) = 0.
- The minimum value occurs at (1, 1) with f(1, 1) = 0.
3. Edge 3: y = 0 and 0 ≤ x ≤ 1
- Substitute y = 0 into the function: f(x, 0) = 4x² - 3(0)² + 2(0) = 4x².
- To find the maximum and minimum values, we need to find the critical points by taking the derivative with respect to x and setting it equal to zero.
- f'(x, 0) = 8x = 0
- Solving for x, we get x = 0.
- Substitute x = 0 into the function: f(0, 0) = 4(0)² - 3(0)² = 0.
- The maximum and minimum values both occur at (0, 0) with f(0, 0) = 0.
4. Edge 4: y = 1 and
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Consider the function f(x, y) = 4x 2 − 3y 2 + 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) = 4x 2 − 3y 2 + 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 JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the function f(x, y) = 4x 2 − 3y 2 + 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 JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f(x, y) = 4x 2 − 3y 2 + 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) = 4x 2 − 3y 2 + 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 JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the function f(x, y) = 4x 2 − 3y 2 + 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 JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f(x, y) = 4x 2 − 3y 2 + 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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