How close does the curve y=sqrt{x} come to the point (3/2,0)?

The distance between (3/2, 0) and y = sqrt(x) is given by D = sqrt((3/2 - x)^2 + (0 - sqrt(x))^2) = sqrt((3/2 - x)^2 + x)
For the distance to be minimum, dD/dx = 0
1/2 * 1/sqrt((3/2 - x)^2 + x) * (-2(3/2 - x) + 1) = 0
1/2 * 1/sqrt((3/2 - x)^2 + x) * (-3 + 2x) = 0
-3 + 2x = 0
x = 3/2

Putting x = 3/2 into the formula for D, we have that D = sqrt(3/2)

Therefore, shortest distance is D = sqrt(3/2)

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