Final answer:
To find the absolute maximum and minimum values of the function f(x, y) on the domain set defined by x² + y² ≤ 1, we must examine both the interior and the boundary of the domain. The absolute maximum and minimum must be found by evaluating the function on the boundary, which can be parametrized using trigonometric functions due to the circular constraint.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x, y) = 2x³y⁴×3 on the set d, where d is defined as the set of all points (x, y) such that x² + y² ≤ 1, we must consider both the interior and the boundary of the set d.
On the interior of the set d, we look for critical points by setting the partial derivatives of f with respect to x and y to zero. However, given that the function includes both x³ and y⁴ and considering the constraint x² + y² ≤ 1 (which represents a unit circle), the critical points in the interior may not yield the absolute extrema.
On the boundary of the set d, we use the constraint x² + y² = 1. Parametrize the boundary using trigonometric functions x = cos(θ), y = sin(θ). Substitute into f to get a function of one variable, f(θ) = 2cos³(θ)sin⁴(θ)×3, defined on [0, 2π]. Then, find the maximum and minimum of this function on the interval to find the absolute extrema of f on the boundary.
The absolute maximum is obtained by finding the highest value of f(θ) within the specified domain and absolute minimum by finding the lowest value of f(θ).