Question

(a) On the axes below, draw level curves for a function f(x, y)

that satisfy the following:
(i) f(x, y) has a saddle point at (3, 2).
(ii) f(x, y) has a local maximum at (0, 1).
(iii) f(x, y) has a gl

Answer 1

Level curves for a function f(x, y) can be drawn by placing a local maximum as closed loops and a saddle point as intersecting hyperbolic curves. The graph should represent the varying heights around these points by using specific (x, y) data pairs and be properly labeled and scaled.

Step-by-step explanation:

Drawing Level Curves for a Function f(x, y)

To draw level curves for a function f(x, y) that satisfy the given conditions, we will consider several points based on the behavior of the function at critical points. We start by placing a saddle point at (3, 2) which means this point should be a part of a curve where the function changes direction from increasing to decreasing or vice versa. A local maximum at (0, 1) would indicate a hill or peak in the function, so level curves around this point should be closed loops with decreasing values as they move away from the point.

Typically, level curves for a local maximum are concentric circles or ellipses, whereas saddle points generally have hyperbolic shapes of level curves intersecting at the saddle point. When sketching these by hand, ensure that the graph is labeled with f(x) and the axes are scaled appropriately. To illustrate the changing values of the function, you could use specific values for (x, y) data pairs to create a varying landscape of height on your graph.