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

$$\mathrm{find}\mathrm{minimum}\mathrm{value}\mathrm{of}$$$$\mid \mathrm{x}-\mathrm{y}\mid +\sqrt{{(\mathrm{x}-3)}^2+{(\mathrm{y}+1)}^2}$$

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

$$\mathrm{the}\mathrm{minimum}\mathrm{value}\mathrm{we}\mathrm{get}\mathrm{when}$$$$\mathrm{x}=\mathrm{y}\Rightarrow \mathrm{let}\mathrm{f}(\mathrm{x},\mathrm{y})=\mid \mathrm{x}-\mathrm{y}\mid +\sqrt{{(\mathrm{x}-3)}^2+{(\mathrm{y}+1)}^2}$$$$\Rightarrow 0+\sqrt{{(\mathrm{x}-3)}^2+{(\mathrm{x}+1)}^2}$$$$\Rightarrow \sqrt{{\mathrm{x}}^2-6\mathrm{x}+9+{\mathrm{x}}^2+2\mathrm{x}+1}$$$$\Rightarrow \sqrt{{2\mathrm{x}}^2-4\mathrm{x}+10}=\sqrt{2({\mathrm{x}}^2-2\mathrm{x}+5)}$$$$=\sqrt{2({(\mathrm{x}-1)}^2+4)}⩾\sqrt{2.4}⩾2\sqrt{2}$$$$\mathrm{minimum}\mathrm{value}2\sqrt{2}\Leftarrow \mathrm{the}\mathrm{answer}$$

Commented by jagoll last updated on 03/Mar/20

$$\mathrm{thank}\mathrm{you}$$

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