Find the point P on the graph of the function y=square root (x) closet to the point (7, 0)

To find the point P on the graph of the function y = \sqrt{x} that is closest to the point (7, 0) , we follow these steps:

1. The distance D between any point (x, \sqrt{x}) on the graph of y = \sqrt{x} and the point (7, 0) is given by the distance formula:

D = \sqrt{(x - 7)^2 + (\sqrt{x} - 0)^2}

2. To find the minimum distance, we minimize the function D^2 to simplify calculations:

D^2 = (x - 7)^2 + x

3. Minimize D^2 with respect to x by taking the derivative of D^2 with respect to x and setting it to zero to find the x -coordinate of P .

4. Once we have x , we can find the corresponding y -coordinate since y = \sqrt{x} .

5. Calculating the x -coordinate of point P gives:
\frac{d(D^2)}{dx} = 2(x - 7) + 1 = 0
2x - 14 + 1 = 0
2x = 13
x = \frac{13}{2}

6. Substitute x = \frac{13}{2} into y = \sqrt{x} to find the corresponding y -coordinate:
y = \sqrt{\frac{13}{2}} = \frac{\sqrt{26}}{2}

So, the point P on the graph of y = \sqrt{x} that is closest to (7, 0) is \left(\frac{13}{2}, \frac{\sqrt{26}}{2}\right) . Therefore, the coordinates simplify to approximately (6.5, 2.5495) .

\boxed{(6.5, 2.5495)}