find-minimum-value-of-x-y-x-3-2-y-1-2-

Question Number 83511 by jagoll last updated on 03/Mar/20

find minimum value of

∣ x - y ∣ + \sqrt{{(x - 3)}^{2} + {(y + 1)}^{2}}

Answered by john santu last updated on 03/Mar/20

the minimum value we get when

x = y \Rightarrow let f (x, y) = ∣ x - y ∣ + \sqrt{{(x - 3)}^{2} + {(y + 1)}^{2}}

\Rightarrow 0 + \sqrt{{(x - 3)}^{2} + {(x + 1)}^{2}}

\Rightarrow \sqrt{x^{2} - 6 x + 9 + x^{2} + 2 x + 1}

\Rightarrow \sqrt{{2 x}^{2} - 4 x + 10} = \sqrt{2 (x^{2} - 2 x + 5)}

= \sqrt{2 ({(x - 1)}^{2} + 4)} ⩾ \sqrt{2.4} ⩾ 2 \sqrt{2}

minimum value 2 \sqrt{2} \Leftarrow the answer

Commented by jagoll last updated on 03/Mar/20

thank you

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