math
posted by Son on .
Find the point on the graph of the function closest to the given point. Function f(x) = xsquared Point (2, 1/2)

using distance from (2,1/2) to y=x^2,
d^2 = (x2)^2 + (y1/2)^2
= (x2)^2 + (x^2  1/2)^2
= x^2  4x + 4 + x^4  x^2 + 1/4
= x^4  4x + 17/4
d = sqrt(x^4  4x + 17/4)
d' = 2(x^3  1)/sqrt(x^4  4x + 17/4)
d' = 0 when x = 1
so, (2,1/2) is closest to (1,1)
______________________
using normal line, we want the line from (2,1/2) normal to the curve. The distance from P to the curve will be minimum along the line normal to the curve.
At (x,y) the slope of x^2 = 2x
so, the normal has slope 1/2x
So, the equation of the normal line from (2,1/2) is
(y  1/2)/(x2) = 1/2x
2x(x^2  1/2) = (x2)
2x^3  x = x + 2
2x^3 = 2
x = 1
So, the normal line from (2,1/2) is the line to (1,1)