Question

Find a point on the curve defined by function y = x^2 that has the minimum distance to the
line defined by y = 2x - 4.
(c) Solve this problem using Lagrange multiplier approach.

Answer 1

When I solved this problem, I ended up with (1, -2), but I think the answer should be (1,1).

Answer 2

Lagrange Equation:
$$
\Lambda(x,y,\lambda) = f(x,y) + \lambda \cdot \Big(g(x,y)-c\Big)
$$
Let \(f(x,y)=\sqrt{\left (x^2\right )^2+(2x-4)^2}\), which is the distance from \(x^2\) to \(2x-4\). We want to minimize this function.

Our constraint is
$$
y=x^2\\
$$
Or
$$
y-x^2=0
$$
Therefore,
$$
\Lambda(x,y,\lambda)=\sqrt{\left (x^2\right )^2+(2x-4)^2}+\lambda \cdot \Big(y-x^2)\Big)
$$
Now let's compute partials:
$$
\Lambda_x={1\over2\sqrt{\Big(x^4+(2x-4)^2\Big )}}\Big(4x^3+8x-16\Big)-2x\lambda\\
\Lambda_y=\lambda\\
\Lambda_\lambda=y-x^2
$$
Setting each equal to zero (you'll need to work out these details):
$$
\Lambda_x=0\\
\Lambda_y=0\\
\Lambda_\lambda=0\\
\implies x\approx1.1795
$$

Hope this helps.