Find the point on the graph x^2 that is closest to the point (4, -3)

Find the point on the graph f(x)=x^2 that is closest to the point (4,-3). Use any necessary derivatives as well as optimization to find the solution(s) by Newton's Method. Explain how you achieved your answer and show that it is accurate. Draw a graph to show that your solution(s) realistically make sense.

well, i understand some parts, i did the distance formula and got sqrt(x-4)^2 + (x^2+3)^2
i took the derivative of that and got 4x^3 +14x - 8
but now i'm stuck with what to do :/
could someone please help?

s² = (x - 4)² + (x² + 3)² ........ distance² from (x,y) on y = x² to (4,-3)
ds/dx = ((x - 4) + (x² + 3) (2x)) / √((x - 4)² + (x² + 3)²)
= (2x³ + 7x - 4) / √((x - 4)² + (x² + 3)²)
2x³ + 7x - 4 = 0 ....... set ds/dx = 0 to find the stationary points
x ≈ 0.52910694 ...... is a min because d²s/dx² > 0

Answer: ≈ (0.52910694, 0.279954159)

It seems like the shortest distance between the point and the curve would be where the segment connecting the point to the curve is perpendicular to the tangent line at that point. So we can assume that the point on the curve is (a, a²). Therefore the gradient from the point (4, -3) to (a, a²) would be: (a² + 3)/(a - 4). The tangent to the curve would have a gradient of 2a at the point (a, a²). So if we wanted the gradients to be perpendicular to each other we would have:

a² + 3....-1
-------- = ---- => 2a³ + 6a = 4 - a => 2a³ + 7a - 4 = 0
a - 4......2a

The solution to this is a = 0.5291 (4dp)

Using Newton's Method we can get:
Xn+1 = Xn - f(Xn) / f ' (Xn) => Xn+1 = Xn - (2(Xn)³ + 7(Xn) - 4)/(4(Xn)² + 7)

Choose Xo = 0.5 and it will converge to 0.529106945...

Therefore, the point that is closest to (4, -3) would be (0.5291, 0.2800).

To test, the distance between these 2 points would be 4.7755 units

The distance from (4, -3) to the origin is 5 units.

The distance from (4, -3) to the point (1, 1) is also 5 units.

Anything beyond these two points would yield distances greater than 5 units so this must be our answer.