To find the point [tex]\( P \)[/tex] on the graph of the function [tex]\( y = \sqrt{x} \)[/tex] that is closest to the point [tex]\( (3, 0) \)[/tex], we need to minimize the distance between any point on the curve [tex]\( (x, \sqrt{x}) \)[/tex] and the point [tex]\( (3, 0) \)[/tex].
Let's denote the distance between a point [tex]\( (x, \sqrt{x}) \)[/tex] on the curve and the point [tex]\( (3, 0) \)[/tex] as [tex]\( D \)[/tex].
First, we write the distance formula [tex]\( D \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
\][/tex]
Substituting [tex]\( (x_1, y_1) = (x, \sqrt{x}) \)[/tex] and [tex]\( (x_2, y_2) = (3, 0) \)[/tex], we get:
[tex]\[
D = \sqrt{(x - 3)^2 + (\sqrt{x} - 0)^2}
\][/tex]
Simplify the expression:
[tex]\[
D = \sqrt{(x - 3)^2 + (\sqrt{x})^2}
\][/tex]
Since [tex]\((\sqrt{x})^2 = x\)[/tex], we can rewrite [tex]\( D \)[/tex] as:
[tex]\[
D = \sqrt{(x - 3)^2 + x}
\][/tex]
To find the minimum distance, we need to find the [tex]\( x \)[/tex] value that minimizes this distance function. Through optimization techniques and calculations, the [tex]\( x \)[/tex]-coordinate of point [tex]\( P \)[/tex] is found to be:
[tex]\[
x = 2.4999999114637803
\][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point [tex]\( P \)[/tex] on the graph of [tex]\( y = \sqrt{x} \)[/tex] that is closest to the point [tex]\( (3, 0) \)[/tex] is approximately:
[tex]\[
x \approx 2.5
\][/tex]
