Calculus Help!!!

Question

Find the point on the parabola x^2 =2y that is closest to the point (4,1).

Answer 1

(2,2) is closest

graph the parabola

visually look at it and see what looks closest to (4,1)

(2,2) looks about right

Answer 2

You have to minimize the distance or equivalently the distance squared between a point (x,y) on the parabola and the point (4,1):

Distance squared: f = (x-4)² + (y-1)²

To make f a function of a single variable, solve the equation of the parabola for y: y = x²/2

Substitute that in f and then it will be f(x), a function of a single variable x.

Then, find the minimum of f(x) using calculus. That will give you the x value at which the minimum occurs. Find the corresponding y value by using y = x²/2