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Mathematics
Calculus
APC.3.1.2 shortest distance between curve and origin
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[QUOTE="HOI, post: 6780887, member: 567247"] My first thought would be to just try each possibility: a) x= 1. The point is (1, 4) which has distance $\sqrt{17}$, about 4.12 from the origin. b) x= 2. The point is (2, 4/\sqrt{2}) which has distance $\sqrt{4+ 8}= \sqrt{12}$, about 3.46, from the origin. c) $x= \sqrt{2}$. The point is $(\sqrt{2}, 4/\sqrt[4]{2})$ which has distance $\sqrt{2+ 16/\sqrt{2}}= \sqrt{2+ 8\sqrt{2}}$, which is about 3.65, from the origin. d) $x= 2\sqrt{2}$. The point is $(2\sqrt{2}, 4/\sqrt[4]{8})$ which has distance $\sqrt{8+ 16/\sqrt{8}}= \sqrt{8+ 8/\sqrt{2}}= \sqrt{8+ 4\sqrt{2}}$, which is about 3.70 from the origin. e) $x= \sqrt[3]{2}$. The point is $\sqrt[3]{2}, 4/\sqrt[6]{2})$ which distance $\sqrt{\sqrt[3]{4}+ 16/\sqrt[3]{2}}$ which is about 3.78 from the origin. Of the four distances, the smallest is 3.46 so (b) x= 2 gives the point closest to the origin! Heavy use of calculator, light use of brain! [/QUOTE]
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APC.3.1.2 shortest distance between curve and origin
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