5 2 Nearest point? What point of the graph of f(x) = 2 ? x is closest to the origin? (Hint: You can minimize the square of the distance.)

Solution Step 1 In this problem we have to find the point in the graph of2 x that is closest to the origin. To find the point we shall make use of the hint. Given f(x) = x 2 2 Let P(x, 2x )be the point of f(x) that is closest to origin. Let D be the square of distance from origin to P. That is D = x + ( x ) 22 25 5 = x + 42( )2x )+x 4 2 25 2 4 = x + 4 5x +x = x 4x +2 25 4 Step 2 Now making use of the hint, to minimize D, put D = 0 4 2 25 We have D = x 4x + 4 D = 4x 8x Now D = 0 3 4x 8x = 0 2 4x(x 2) = 0 4x = 0 or x 2 = 0 x = 0or x = 2 x = 0or x = ± 2