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Question Number 141475 by bramlexs22 last updated on 19/May/21

$$Findthesmallestvalueof$$ $$\sqrt{x^2+y^2}amongallvaluesof$$ $$x\mathrm{\&}ysatisfying3x-y=20$$

Answered by MJS_new last updated on 19/May/21

$$y=3x-20$$ $$f\left(x\right)=\sqrt{10(x^2-12x+40)}$$ $$x=t+6$$ $$f\left(t\right)=\sqrt{10(t^2+4)}$$ $$\min \left(f\left(t\right)\right)=2\sqrt{10}$$

Answered by mathmax by abdo last updated on 19/May/21

$$\phi \left(\mathrm{x}\right)=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}\mathrm{and}\mathrm{y}=3\mathrm{x}-20\Rightarrow \phi \left(\mathrm{x}\right)=\sqrt{{\mathrm{x}}^2+{(3\mathrm{x}-20)}^2}$$ $$=\sqrt{{\mathrm{x}}^2+{9\mathrm{x}}^2-120\mathrm{x}+400}=\sqrt{{10\mathrm{x}}^2-120\mathrm{x}+400}$$ $${\phi }^{{}^{\prime }}\left(\mathrm{x}\right)=\frac{20\mathrm{x}-120}{2\sqrt{{10\mathrm{x}}^2-120\mathrm{x}}+400}=\frac{10\mathrm{x}-60}{\sqrt{{10\mathrm{x}}^2-120\mathrm{x}+400}}=\frac{10(\mathrm{x}-6)}{\sqrt{....}}$$ $${\phi }^{{}^{\prime }}\left(\mathrm{x}\right)=0\Rightarrow \mathrm{x}=6\mathrm{and}{\phi }^{{}^{\prime }}\left(\mathrm{x}\right)>0\Rightarrow \mathrm{x}>6$$ $${{\phi }^{{}^{\prime }}\left(\mathrm{x}\right)<0\Rightarrow \mathrm{x}<6\Rightarrow \inf \phi \left(\mathrm{x}\right)=\phi \left(6\right)=\sqrt{6^2+(18-20})}^2=\sqrt{36+4}$$ $$=\sqrt{40}=2\sqrt{10}$$

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