Raymond B. answered • 04/12/20
Math, microeconomics or criminal justice
First guess would be almost directly to the right, and left about 3 units
about (sqr8.5, 8.5) rounded off to one decimal place
Visually, it's probably slightly below y=9 and slightly less than x=3
also about (-3, 9)
slightly less than 3 and 9
Connect the point (0,9) to the nearest point on the parabola. It's slope is the negative inverse of the parabola's slope where they intersect. The slope of f(x) = x^2 is 2x, the derivative. It's inverse negative is -1/2x The line connecting (0,9) to the closest point on the parabola is a straight line with slope perpendicular to the tangent to the parabola at that closest point.
use the point slope formula to get the equation of the line connecting (0,9) to the closest point (x,y) on the parabola: y= (-1/2x)(x) + 9 or y=-1/2 + 9 = 8.5
That's the y coordinate of the closest point. The x coordinate of the closest point is the square root of 8.5, both positive & negative square roots: (sqr8.5, 8.5) and (-sqr8.5, 8.5)
sqr8.5 = square root of 8.5 which is about 2.91 or -2.91
The method is to find the equation for the line from (0,9) to the closest point on the parabola, and then set the two equations equal. Set the line equation = the parabola equation
y=(-1/2x)x + 9 is the line equation
y=x^2 is the parabola equation
set them equal, solve for x. square x you have y
(x,y) is the closest point. There are two points equidistant to (0,9) both as close as it gets, slightly less than 3
at (3,9) the distance is exactly 3 to (0,9). That line is not perpendicular to the tangent line at the parabola point (3,9). To be closest, the line has to be perpendicular to the tangent line, the slope of the parabola.
