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If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.?
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If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3...
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If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3...
Approach:
To find the largest value which x can have, we can use the concept of inequalities to optimize the value of x.
Given:
x + y + z = 5
xy + yz + xz = 3
Using AM-GM inequality:
From the given equations, we have:
(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + xz)
Using the values from the given equations:
25 = x^2 + y^2 + z^2 + 2(3)
25 = x^2 + y^2 + z^2 + 6
x^2 + y^2 + z^2 = 19
Now, using the AM-GM inequality, we know that:
(x^2 + y^2 + z^2)/3 ≥ (xyz)^(2/3)
19/3 ≥ (xyz)^(2/3)
8.67 ≥ (xyz)^(2/3)
Maximizing x:
To maximize x, we need to maximize the value of xyz. Since we want to find the largest value of x, we can assume y = z.
So, x + 2y = 5
Let's assume y = z = t
x + 2t = 5
x = 5 - 2t
Substitute back in the inequality:
(xyz)^(2/3) = ((5 - 2t)t^2)^(2/3) = (5t^2 - 2t^3)^(2/3)
Now, differentiate the above expression with respect to t and equate it to 0 to find the maximum value of x.
By solving the derivative, we can find the maximum value of x.
To find the largest value which x can have, we can use the concept of inequalities to optimize the value of x.
Given:
x + y + z = 5
xy + yz + xz = 3
Using AM-GM inequality:
From the given equations, we have:
(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + xz)
Using the values from the given equations:
25 = x^2 + y^2 + z^2 + 2(3)
25 = x^2 + y^2 + z^2 + 6
x^2 + y^2 + z^2 = 19
Now, using the AM-GM inequality, we know that:
(x^2 + y^2 + z^2)/3 ≥ (xyz)^(2/3)
19/3 ≥ (xyz)^(2/3)
8.67 ≥ (xyz)^(2/3)
Maximizing x:
To maximize x, we need to maximize the value of xyz. Since we want to find the largest value of x, we can assume y = z.
So, x + 2y = 5
Let's assume y = z = t
x + 2t = 5
x = 5 - 2t
Substitute back in the inequality:
(xyz)^(2/3) = ((5 - 2t)t^2)^(2/3) = (5t^2 - 2t^3)^(2/3)
Now, differentiate the above expression with respect to t and equate it to 0 to find the maximum value of x.
By solving the derivative, we can find the maximum value of x.
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If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.?
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If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.?.
If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x , y and z are real numbers such that x+ y +z=5 , and xy+ yz+ xz=3, what is the largest value which x can have? P.s. try to solve using inequalities please rather than using the number system method.?.
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