How do you find the point on the the graph y=sqrtx which is plosest to the point (4,0)?

ceriserasb6

ceriserasb6

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2022-08-21

How do you find the point on the the graph y = x which is plosest to the point (4,0)?

Answer & Explanation

Bryanna Villarreal

Bryanna Villarreal

Beginner2022-08-22Added 5 answers

First note that the distance between an arbitrary point ( x , y ) = ( x , x ) on the graph of y = x and the point (4,0) is
( x - 4 ) 2 + ( x - 0 ) 2 = x 2 - 7 x + 16
Next, note that the value of x where d ( x ) = x 2 - 7 x + 16 is minimized is the same value of x where s ( x ) = ( d ( x ) ) 2 = x 2 - 7 x + 16 is minimized.
Since s ( x ) = 2 x - 7 , s ( x ) and d(x) are minimized at x = 7 2 = 3.5 (note that s ( x ) = 2 > 0 so that the critical point is a minimum).
Therefore, the point on the graph of y = x that is closest to (4,0) is
( 7 2 , 7 2 ) ( 3.5 , 1.87083 ) . The minimum distance is
d ( 7 2 ) = 49 4 - 49 2 + 16 = 15 4 = 15 2 1.93649

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