BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 40RE
To determine

Tofind:the point on hyperbola which is closest to the point given below.

Expert Solution

Answer to Problem 40RE

Thepoint which is closest to (3,0) is (4,84) or (4,2)

Explanation of Solution

Given:

  xy=8 .

Closest point is (3,0) .

Concept used:

Distance formula:

  d=(xx1)2+(yy1)2 .

Local minima Occurs when f(x) changes its sign from negative to positive.

Calculation:

Every point on the hyperbola xy=8 is in the form of (x,8x) .

Distance between the point (x,8x) and (3,0) is

  d(x)=(x-3)2+(8x-0)2 .

Her to find the x foe which the d(x) id minimum.

That is

  f(x)=x2-6x+9+64x2 .

  f(x)=2x-6-128x3

The graph of the f(x) is shown below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4, Problem 40RE

Local minima Occurs when f(x) changes its sign from negative to positive.

Therefore, the graph above,shows that the local minima occurs when x=4 .

Since there is only one local Minima and limx±f(x)= .

Hence the point which is closest to (3,0) is (4,84) or (4,2) .

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