If x, y and z are real numbers such that x + y + z = 5 and xy + yz + zx = 3, what is the largest value that x can have?a)5/3b)13/3c)andradic;19d)None of theseCorrect answer is option 'B'. Can you explain this answer?

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If x, y and z are real numbers such that x + y + z = 5 and xy + yz + zx = 3, what is the largest value that x can have?

a)
5/3
b)
13/3
c)
√19
d)
None of these

Correct answer is option 'B'. Can you explain this answer?

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Verified Answer

If x, y and z are real numbers such that x + y + z = 5 and xy + yz + z...

The given equations are x + y + z = 5 — (1) , xy + yz + zx = 3 — (2)

xy + yz + zx = 3

x(y + z) + yz = 3

⇒ x ( 5 - x ) +y ( 5 – x – y) = 3

⇒ -y² - y(5 - x) - x² + 5x = 3
⇒ y² + y(x - 5) + (x²- 5x + 3) = 0
The above equation should have real roots for y, => Determinant >= 0

⇒ b² - 4ac0

⇒ (x - 5)² - 4(x² - 5x + 3) ≥ 0

⇒ 3x² - 10x - 13 ≤ 0

⇒ -1 ≤ x ≤ 13/3

Hence maximum value x can take is 13/3, and the corresponding values for y,z are 1/3, 1/3

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Most Upvoted Answer

If x, y and z are real numbers such that x + y + z = 5 and xy + yz + z...

Let's call the three numbers x, y, and z. We are given that x+y+z=5 and xy+yz+zx=3.

We can rewrite the first equation as x+y+z=5 as xy+yz+zx+2xy+2yz+2zx=3+2xy+2yz+2zx.

Adding the two equations together, we have 3+2xy+2yz+2zx=3+3+2xy+2yz+2zx.

Simplifying, we have 6=6+2xy+2yz+2zx.

Subtracting 6 from both sides, we have 0=2xy+2yz+2zx.

Dividing both sides by 2, we have 0=xy+yz+zx.

We can rewrite this equation as x(y+z)+yz=0.

Since x, y, and z are real numbers, we know that y+z and yz must have opposite signs (if they had the same sign, then their sum would not be zero).

This means that either y and z are both negative, or one of them is negative and the other is positive.

If both y and z are negative, then y+z is negative. Since the sum of three real numbers is 5, this would mean that x is positive.

However, if one of y or z is negative and the other is positive, then their sum y+z is positive. Since the sum of three real numbers is 5, this would mean that x is negative.

Therefore, x cannot be positive.

The largest value that x can have is 0.

Therefore, the answer is c) 0.

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If x, y and z are real numbers such that x + y + z = 5 and xy + yz + zx = 3, what is the largest value that x can have?a)5/3b)13/3c)√19d)None of theseCorrect answer is option 'B'. Can you explain this answer?

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