To find the point on the curve y = √x that is closest to the point (3, 0), we can use the distance formula. The distance d between a point on the curve, given by (x, √x), and the point (3, 0) can be expressed as:
d = √[(x – 3)² + (√x – 0)²]
For simplicity, we can minimize the square of the distance instead, which is:
d² = (x – 3)² + (√x)²
Substituting √x gives us:
d² = (x – 3)² + x
Expanding this, we get:
d² = (x² – 6x + 9) + x = x² – 5x + 9
To minimize d², we can take its derivative and set it to zero:
f(x) = x² – 5x + 9
f'(x) = 2x – 5
Setting the derivative equal to zero:
2x – 5 = 0
Solving for x gives:
x = 2.5
Now, we can find the corresponding y-coordinate on the curve:
y = √(2.5) = √(5/2) ≈ 1.581
Thus, the point on the curve y = √x that is closest to (3, 0) is approximately (2.5, 1.581).