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Minimizing rectangle perimeters All rectangles with an area of 64 have a perimeter given by $P(x)=2 x+\frac{128}{x},$ where $x$ is the length of one side of the rec...

Question

Minimizing rectangle perimeters All rectangles with an area of 64 have a perimeter given by $P(x)=2 x+\frac{128}{x},$ where $x$ is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function on the interval $(0, \infty) .$ What are the dimensions of the rectangle with minimum perimeter?

Minimizing rectangle perimeters All rectangles with an area of 64 have a perimeter given by $P(x)=2 x+\frac{128}{x},$ where $x$ is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function on the interval $(0, \infty) .$ What are the dimensions of the rectangle with minimum perimeter?



Answers

Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter $p$ is a square.


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