Final answer:
The function f(x, y) = 1/5x² + 8y² has a domain that includes all points in the xy-plane, a range of all real numbers greater than or equal to 0, and level curves which are ellipses of the form 5x² + 8y² = c.
Step-by-step explanation:
The function provided is f(x, y) = ⅓x² + 8y². To determine the correct choice regarding the domain, range, and level curves, the function must be analyzed.
The domain of a function is the set of all possible input values. Since the function is composed of squared terms, it can accept any real number for both x and y without causing any undefined behavior. Thus, the domain includes all points in the xy-plane.
Considering the range, since the terms are squared, their smallest possible value is 0, occurring when both x and y are 0. Because the coefficients are positive, the output of the function will be greater than or equal to 0. Thus, the range of the function is all real numbers greater than or equal to 0.
The level curves are formed when the function equals a constant value c. In this case, setting f(x, y) to c gives us the equation 5x² + 8y² = c. As c takes on different positive values, this equation represents a series of ellipses centered at the origin.
With this analysis, it's clear that the correct option is:
- Domain: all points in the xy-plane
- Range: real numbers >= 0
- Level curves: ellipses 5x² + 8y² = c