All rectangles with an area of 64 have a perimeter given by...

All rectangles with an area of 64 have a perimeter given by $P(x)=2 x+128 / x,$ where $x$ is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function. What are the dimensions of the rectangle with minimum perimeter?

Key Concepts

Geometric Constraints in Rectangular Dimensions

In rectangle optimization problems, the constraint imposed by a fixed area creates a relationship between the length and width of the rectangle. This relationship allows the perimeter to be expressed as a function of a single variable. Understanding how to manipulate and apply this constraint is key to reducing the problem to a manageable single-variable calculus problem, enabling the application of derivative-based optimization techniques.

Optimization

Optimization involves finding the maximum or minimum values of a function subject to certain constraints. In problems involving geometric figures, such as rectangles with a fixed area, the goal is to determine the dimensions that yield the smallest or largest possible perimeter, area, or other attributes. This requires formulating a function that describes the quantity to be optimized and then applying calculus-based techniques to find its extreme values.

Derivative and Critical Point Analysis

The use of derivatives is fundamental in optimization. By taking the derivative of the function that models the problem (in this case, the perimeter function) and setting it equal to zero, we can locate critical points where the function may attain a minimum or maximum. This method of finding critical points is standard in problems where the relationship between variables is continuous and differentiable.

Second Derivative Test

After identifying potential critical points via the first derivative, the second derivative test is applied to determine whether these points correspond to minima or maxima. If the second derivative at a critical point is positive, the function is locally convex and the point is a local minimum. This test is essential to confirm that the critical point found indeed yields the minimal perimeter.

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