Calculus question about finding x of P

Question

Find the point P on the graph of the function y=√x closest to the point (9,0) The x coordinate of P is

Answer 1

Let the point on the curve of y = √x that is closest to (9, 0) be the point (a, b)

The distance between (a, b) and (9, 0) is given by the distance formula:

d= √[(9 - a)² + (0 - b)²] but we also know that b = √a making the distance equation a function of a:

d(a) = √[(9 - a)² + (0 - √a)²]

d(a) = √[(9 - a)² + (-√a)²]

d(a) = √[81 - 18a + a² + a]

d(a) = √(a² - 17a + 81)

To minimize, take the derivative and set it equal to zero:

d'(a) =1/2(a² - 17a + 81)^-1/2(2a - 17)

(2a - 17)/(2√(a² - 17a + 81)) = 0 when 2a - 17 = 0

2a - 17 = 0

2a = 17

a = 17/2

b = √(17/2) = √(34)/2

So the x coordinate of the point is 17/2 or 8.5