anonymous
  • anonymous
Optimization. Find the minimum distance of a point on the graph xy^2=16 from the origin
Mathematics
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anonymous
  • anonymous
Optimization. Find the minimum distance of a point on the graph xy^2=16 from the origin
Mathematics
chestercat
  • chestercat
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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Xishem
  • Xishem
This is your constraint (rewritten): \[f(x)=\sqrt{\frac{16}{x}}\]Your objective function (the value you are trying to minimize) is the distance between some point on the graph and (0,0):\[D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(0-x)^2+(0-\sqrt{\frac{16}{x}})^2}=\sqrt{(-x)^2+\frac{16}{x}}\]It's enough to minimize the value under the radical, and since it makes the problem a bit simpler, let's go ahead make your O.F all the stuff underneath the radical: \[D=x^2+\frac{16}{x}\]This is the function we are trying to minimize, so let's go ahead and differentiate it: \[\frac{dD}{dx}=2x-\frac{16}{x^2}\]Now we need to find the critical numbers.\[0=2x-\frac{16}{x^2}\rightarrow 2x=\frac{16}{x^2} \rightarrow 2x^3=16 \rightarrow x^3=8 \rightarrow x=2\]This means that the point that is closest to (0,0) is (x,f(x))\[f(x)=\sqrt{\frac{16}{x}} \rightarrow f(2)=\sqrt{\frac{16}{2}}=\sqrt{8}\approx 2.8284\]The point closest to (0,0) in the function xy^2=16 is (2,2.8284).

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