Question

Find a function of two variables f(x,y) for which D = 0, but the function has a minimum. Exclude the trivial functions f(x,y) = constant. Explain why your example has a minimum and show in detail that D = 0 at that minimum.

Answer 1

Answer:

Example 13.8.1

Finding critical points and relative extrema

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Let f(x,y)=x2+y2−xy−x−2. Find the relative extrema of f.

SolutionWe start by computing the partial derivatives of f:

fx(x,y)=2x−y−1  and  fy(x,y)=2y−x.

Each is never undefined. A critical point occurs when fx and fy are simultaneously 0, leading us to solve the following system of linear equations:

2x−y−1=0  and  −x+2y=0.

This solution to this system is x=2/3, y=1/3. (Check that at (2/3,1/3), both fx and fy are 0.

The graph in Figure 13.8.1 shows f along with this critical point. It is clear from the graph that this is a relative minimum; further consideration of the function shows that this is actually the absolute minimum.