Question 382720
We want to minimize the distance between the points (x, x^2) and (3,1). Using the distance formula, this distance z is equal to

{{{z = sqrt ((x-3)^2 + (x^2 - 1)^2) = sqrt(x^4 - x^2 - 6x + 10)}}}

We wish to minimize the value of z. By taking the derivative of z and setting it to zero, we obtain

{{{dz/dx = (4x^3 - 2x - 6)/(2*sqrt(x^4 - x^2 - 6x + 10)) = (2x^3 - x - 3)/(sqrt(x^4 - x^2 - 6x + 10)) = 0}}}

This occurs when

{{{2x^3 - x - 3 = 0}}} and the denominator is nonzero.

I had to use a calculator to determine that the only real root is x = 1.289623 (meaning that a critical point is located there, it can be checked that it minimizes the distance). Therefore the closest point on the graph y = x^2 to (3,1) is (1.289623, 1.663127)