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Differentiation is a mathematical process used to determine how a function changes as its input changes. In this exercise, we are tasked with finding a point on a curve that minimizes the perimeter of a particular rectangle. We begin by expressing the perimeter of the rectangle using the function of the curve. The curve is given as \( y = x^{-2} \), and thus, the perimeter \( P \) is expressed as \( P = 2x + 2x^{-2} \).

Finding the minimum perimeter requires us to differentiate this expression with respect to \( x \). By performing differentiation on \( P \), using the power rule of differentiation, we calculate the first derivative: \( \frac{dP}{dx} = 2 - 4x^{-3} \).

This derivative represents the rate of change of the perimeter concerning changes in \( x \). By setting this derivative to zero, we can find the points where this rate of change is zero, leading us to critical points where a minimum may occur.

Critical points are values of the input, \( x \), where the first derivative of a function equals zero or is undefined. Such points often correspond to the peaks, troughs, or flat areas of a curve, making them crucial in optimization problems.

In this exercise, after differentiating the perimeter function, we find the critical points by solving \( 2 - 4x^{-3} = 0 \).

Solving this equation involves rearranging terms to get \( 4x^{-3} = 2 \), which simplifies to \( x^3 = 2 \), leading us to \( x = \sqrt[3]{2} \).

This critical point is pivotal because it indicates a potential location where the perimeter is at its minimum value. However, we have to confirm whether this point is indeed a minimum, which is where the second derivative test comes into play.

The second derivative test is a tool used to determine the concavity of a function at a given critical point, which in turn indicates whether the critical point is a local minimum or maximum. For the problem at hand, we apply this test to verify that the critical point \( x = \sqrt[3]{2} \) results in the minimum perimeter.

We compute the second derivative of the perimeter function: \( \frac{d^2P}{dx^2} = 12x^{-4} \).

Evaluating this expression shows \( \frac{d^2P}{dx^2} > 0 \) for all \( x > 0 \), suggesting that the function is concave up everywhere in the first quadrant. Thus, the point \( x = \sqrt[3]{2} \) is confirmed as a point of local minimum.

Finally, after confirming the location of the minimum, we calculate \( y \) using \( y = x^{-2} \) at \( x = \sqrt[3]{2} \), resulting in \( y = 2^{-2/3} \). The coordinates of the point \( P \) that yields the smallest perimeter are \( P = (\sqrt[3]{2}, 2^{-2/3}) \).