How do you find points given the following?

Question

Find the points on the graph of the function that are closest to the given point.
f(x) = x^2,(0, 9)

Answer 1

A  point on the curve y = x² can be expressed as (x, x²).

 

Minimize D = distance from (x, x²) to (0,9):

 

     D = √[(x-0)² + (x²-9)²]

 

        = √[x² + x⁴ - 18x²+ 81]

 

        = √[x⁴ - 17x² + 81]

 

D is minimized when  S = x⁴ - 17x² + 81 is minimized.

 

      S' = 4x³ - 34x = 0

 

             2x(2x² - 17) = 0

 

              x = 0 or x = ±√(17/2)

 

If x < -√(17/2), then S' < 0.  So, S is decreasing.

 

If -√(17/2) < x < 0, then S' > 0.  So, S is increasing.

 

If 0 < x < √(17/2), then S' < 0.  So, S is decreasing.

 

If x > √(17/2), then S' > 0.  So, S is increasing.

 

S is an even function, so the graph is symmetric with respect to the y-axis.

 

Minima when x = ±√(17/2)

 

Points on the graph of y = x² that are closest to (0,9) are:

 

    (±√(17/2), 17/2) ≈ (±2.915476, 8.5)