Final answer:
Using Lagrange multipliers to maximize the area of a rectangle inscribed in an ellipse results in finding that the maximum area occurs when the rectangle's sides are proportional to the ellipse's axes. For a general ellipse, this does not form a square unless the ellipse is actually a circle. Hence, the maximum area of the rectangle inscribed within an ellipse is achieved when the rectangle is a square, which is a special case.
Step-by-step explanation:
To find the maximum area of a rectangle inscribed in an ellipse using Lagrange multipliers, one can set up the problem as follows:
- Let the semi-axes of the ellipse be given by a and b and the equation of the ellipse be given by x^2/a^2 + y^2/b^2 = 1.
- Assume that a corner of the rectangle is at point (x,y) in the first quadrant.
- The area of the rectangle is then given by A = 4xy since there will be four such corners, by symmetry.
- This is subject to the constraint x^2/a^2 + y^2/b^2 = 1.
To maximize A given this constraint, we set up the Lagrange function L(x, y, λ) = 4xy - λ(x^2/a^2 + y^2/b^2 - 1), and find the partial derivatives of L with respect to x, y, and λ, and set them equal to zero.
After solving the system of equations, we find that the maximum area occurs when x/a = y/b, which implies that the rectangle's sides are proportional to the ellipse's axes. Since for ellipses with different axes lengths (non-circles), the maximized rectangle is not a square, the maximum area is achieved when the rectangle is a square only if the ellipse is a circle (a = b).
The correct answer to the problem is (d) The maximum area is achieved when the rectangle is a square, which holds true for the special case where the ellipse is also a circle.