Final Answer:
The level curves of the function g(x, y) = x² y are hyperbolic.
Step-by-step explanation:
The function g(x, y) = x² y represents a family of curves in the xy-plane where each curve has the same value of g for different x and y values. To find these level curves, we set g(x, y) equal to a constant, say k, and solve for y in terms of x to sketch these curves. Thus, for g(x, y) = k, we have y = k / x².
The level curves, as determined by the equation y = k / x², are hyperbolic. As x approaches zero, the curves approach the y-axis asymptotically. As x increases or decreases, the curves extend in both positive and negative y-directions, forming hyperbolas that open up or down based on the sign of k.
These hyperbolic curves illustrate the relationship between x and y that yields a constant value for g(x, y) = x² y. The nature of hyperbolas, converging towards the y-axis and extending in the y-direction, emphasizes how changes in x and y influence the function's output, maintaining the same g-value across each individual curve in the family of level curves.