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

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    using distance from (2,1/2) to y=x^2,

    d^2 = (x-2)^2 + (y-1/2)^2
    = (x-2)^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)/(x-2) = -1/2x

    2x(x^2 - 1/2) = -(x-2)
    2x^3 - x = -x + 2
    2x^3 = 2
    x = 1

    So, the normal line from (2,1/2) is the line to (1,1)

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