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Home Continuous Functions Maxima and Minima Examples Example 4
 
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Example 4

Find the point on the line y = 2x + 3 which is at minimum distance from the origin. The distance is given by

03_continuous_functions-154.gif

and substituting 2x + 3 for y,

03_continuous_functions-155.gif

This is defined on the whole real line.

Step 1

03_continuous_functions-156.gif

Step 2

03_continuous_functions-157.gif = 0

only when

5x + 6 = 0, or x = -(6/5)

Step 3

03_continuous_functions-158.gif

At x = -(6/5), 5x + 6 = 0 and z > 0 so

d2z/dx2 = 5/z > 0.

By the Second Derivative Test, z has a minimum at x = -(6/5).

CONCLUSION

The distance is a minimum at

x = -(6/5), y = 2x + 3 = 3/5.

The minimum distance is

03_continuous_functions-159.gif.

This is shown in Figure 3.5.9.

03_continuous_functions-160.gif

Figure 3.5.9
Home Continuous Functions Maxima and Minima Examples Example 4

Last Update: 2006-11-25