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If x + y + z = 5 and xy + yz + zx = 3, then least and largest value of x are

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$\frac{10}{3}, 5$

$- 1, \frac{13}{3}$

$\frac{17}{3}, 7$

$20, \frac{5}{3}$

answer is B.

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Detailed Solution

$\begin{aligned} z = 5 - x - y \\ xy + z (y + x) = 3 \\ \Rightarrow xy + (y + x) (5 - (x + y)) = 3 \\ \Rightarrow x^{2} + y^{2} + xy - 5 x - 5 y + 3 = 0 \end{aligned}$
as quadratic in $y then Δ \geq 0$

$x = - 1, \frac{13}{3}$

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