1. ## Optimization Problem

Problem: Which point on the parabola $\displaystyle y = x^2$ is nearest to (4, 0)? Find the coordinates to two decimals. (Hint: Minimize the square of the distance to avoid square roots.)

What I've Done:
I drew a picture to try and help me solve it, and tried to make a triangle between the curve $\displaystyle x^2$ and
the point (4 , 0). I just don't really know how to get any equations for the problem.

Help? I'd really appreciate it!

2. Originally Posted by lysserloo
Problem: Which point on the parabola $\displaystyle y = x^2$ is nearest to (4, 0)? Find the coordinates to two decimals. (Hint: Minimize the square of the distance to avoid square roots.)

What I've Done:
I drew a picture to try and help me solve it, and tried to make a triangle between the curve $\displaystyle x^2$ and
the point (4 , 0). I just don't really know how to get any equations for the problem.

Help? I'd really appreciate it!
A general point on the curve $\displaystyle y=x^2$ is $\displaystyle (x,x^2)$, and the square distance from this point to $\displaystyle (4,0)$ is:

$\displaystyle D^2=(x-4)^2+(x^4)$

Now we want to find the $\displaystyle x$ that minimises $\displaystyle D^2$ and we do this by solving $\displaystyle (D^2)'=0$

Which leaves you with the equation:

$\displaystyle 2x^3+x-4=0$

which has one positive root and no negative roots (Descartes rule of signs tell us this), and this positive root is of necessity a minimum for $\displaystyle D^2$.

You are expected to find this root numerically.

CB