What is the point on the curve y = √x that is closest to the point (3, 0)?

To find the point on the curve y = √x that is closest to the point (3, 0), we will use the concept of distance in coordinate geometry.

First, express the distance d from the point (3, 0) to any point (x, y) on the curve. Since y = √x, we can rewrite the distance as:

d = √((x - 3)² + (√x - 0)²)

Squaring the distance to make it simpler, we have:

d² = (x - 3)² + (√x)²

Substituting y = √x into the equation gives:

d² = (x - 3)² + x

Expanding this, we get:

d² = (x² - 6x + 9) + x = x² - 5x + 9

To minimize the distance, we can take the derivative of d² with respect to x and set it equal to zero:

d(d²)/dx = 2x - 5 = 0

Solve for x:

2x = 5

x = 2.5

Now substitute x = 2.5 back into the equation of the curve to find y:

y = √(2.5) = √(25/10) = √2.5

Now, we have the coordinates of the point on the curve that is closest to (3, 0):

(2.5, √2.5)

Therefore, the point on the curve y = √x that is closest to the point (3, 0) is approximately:

(2.5, 1.581)

In summary, the closest point on the curve y = √x to the point (3, 0) is (2.5, √2.5) or approximately (2.5, 1.581).