   Chapter 14.1, Problem 79E

Chapter
Section
Textbook Problem

Use a computer to investigate the family of surfaces z = x2 + y2 + cxy. In particular, you should determine the transitional values of c for which the surface changes from one type of quadric surface to another.

To determine

To investigate: The family of the surfaces z=x2+y2+cxy use it to describe the shape of the graph depend on c .

Explanation

The given surface is z=x2+y2+cxy .

Graph at c=0 :

Consider the values c=0 and draw the graph of the surface z=x2+y2+cxy as follows,

Using online graphing calculator and draw the graph of z=x2+y2 as shown below in Figure 1.

From Figure 1, it can be observed that the shape is elliptic paraboloid at the origin and the valley point appears at origin.

Thus, it has local minimum at origin.

Graph at c=10 :

Consider the values c=10 and draw the graph of the surface z=x2+y210xy as follows,

Using online graphing calculator and draw the graph of z=x2+y210xy as shown below in Figure 2.

From Figure 2, it can be observed that the shape is hyperbolic paraboloid and the valley point appears approximately at the origin.

Thus, it has local minimum at origin

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