Final answer:
The global extreme values of the function are found by examining the gradient for critical points, evaluating the function at the corner points of the domain, and along the boundaries. The domain restrictions form a triangle within the first quadrant to be investigated for these values.
Step-by-step explanation:
To determine the global extreme values of the function f(x, y) = 4x³ + 4x²y + 2y² on the given domain where x, y ≥ 0 and x + y ≤ 1, we first need to understand that since the domain restricts x and y to be non-negative and with their sum not exceeding 1, the function's behavior is contained within a triangle in the first quadrant of the xy-plane.
Global extremes can occur at interior points where the gradient is zero or on boundary points. Since the function is polynomial and smooth, we can check for critical points by setting the partial derivatives equal to zero:
- ∂f/∂x = 12x² + 8xy = 0
- ∂f/∂y = 4x² + 4y = 0
However, within the domain constraints, these can only be satisfied simultaneously when x = y = 0, which is a point we must consider alongside the boundaries. The boundaries are formed by the lines y = 0, x = 0, and x + y = 1. We must examine the function along these boundaries and at corner points (0,0), (1,0), (0,1).
A thorough examination of the function along the edges of this domain and at the corner points will yield the global maximum and minimum. Factoring and simplifying the polynomial along the boundary lines, or evaluating it at the corner points, are both effective methods for finding these extremal values.