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.

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.

anonymous · Answer

When I tried solving this problem, I ended up with (1, -2), but I think the correct answer is (1,1).

freckles · Answer

You used the distance formula?

anonymous · Answer

Yes

anonymous · Answer

When I used the Lagrange method, I got something different. Not sure how I made an error.

freckles · Answer

Did you have something like this:
$$f(x_1,x_2)=(x_1-x_2)^2+(x^2_1-2x_2+4)^2 $$
then we have 
$$f_{x_1}=\lambda g_{x_1 } \ f_{x_2}=\lambda g_{x_2}$$
but I'm having trouble on what the constraint is ... Can I see what you did?

anonymous · Answer

x^2 -y  - lambda(2x-y-4) = F(x,y, lambda)
x^2 - y -2lambda x + lamda y + 4 lamda

partial x: 2x - 2lambda = 0
partial y: -1 + lamda = 0
partial lamda: -2x + y + 4 = 0.
And then I just solved.

freckles · Answer

oh I think you just got it backwards

freckles · Answer

$$2x-y-4-\lambda(x^2-y)=F \ F_x=2-2x \lambda=0 \ F_y=-1+\lambda=0 \ F_\lambda=-(x^2-y)=0$$

freckles · Answer

that works