Using the first derivative find a point on the line that is closest to point P(2,-2). Set up using the distance formula. Express answer using one decimal place. y=3x+14

Answer 1

Distance square f(x) = (x - 2)² + (3x + 16)²; f'(x) = 2(x - 2) + 6(3x + 16) = 0; 20x + 92 = 0; x = - 4.6;

y = 3( - 4.6) + 14 = 0.2.

f''(x) = 20 > 0. So, we have minimum distance.

Answer: point (-4.6, 0.2) is closest point of line y = 3x + 14 from point (2, -2)

Answer 2

y = 3x +14

(2,-2) is closest to what point on that line

that point is on a perpendicular line with slope = -1/3

y = -x/3 + b

-2 = -2/3 +b

b = -2 + 2/3 = -4/3

y = -x/3 - 4/3 intersects y=3x + 14 where

3x + 14 = -x/3 -4/3

9x + 42 = -x - 4

10x = - 46

x = -4.6

y = 3(-4.6) +14 = -13.8 + 14 = 0.2

(-4.6, 0.2) is the point on the line closest to (2, -2)

it might help to graph the lines and points