Final answer:
The question is about optimizing a function representing the sum of squares of areas of four smaller rectangles formed by cutting a larger rectangle with length l and width w. Calculus or a geometric approach can be used for finding the maximum and minimum values, which involve setting partial derivatives to zero or utilizing properties of squares, respectively.
Step-by-step explanation:
The question involves finding the maximum and minimum values of the sum of squares of the areas of the four smaller rectangles formed when a rectangle with length l and width w is cut by two lines parallel to its sides. To solve this, we can assign variables to the lengths of the segments into which l and w are divided, say a and b for the lengths and c and d for the widths, such that a + b = l and c + d = w. Then, we calculate the areas of the smaller rectangles, square these areas, and add them up to form a function that depends on a and c (since b and d are determined by a and c).
This function can be optimized using calculus or by recognizing that it is a sum of squared terms, and thus convex, allowing us to consider boundary values for the optimums. After finding the function, we can set its partial derivatives with respect to a and c to zero to find critical points, which could give the minimum or maximum values.
Alternatively, we might observe that for a fixed perimeter, a square has the maximum area, and this principle could guide us in finding the answer without calculus.
However, without explicit functions and constraints, we cannot provide the exact maximum and minimum values. It's also worth noting that if the cut lines are equally spaced, then the four smaller rectangles will have equal areas, and this would minimize the sum of the squares of the areas.