Finding minimum distance between two points on two different parabolas

[smashX]

Hi everyone. I've got a homework problem with Differential Calculus that drives me crazy for a couple of days so far and would like to ask you guys for a few suggestions

Homework Statement

Given two parabolas, (C₁): y = x² and (C₂): y = - (4-x)². Find two points, each point on one parabola so that the distance between them is smallest.

Homework Equations

The distance equation (I think)

The Attempt at a Solution

Well, I called the point on (C₁) (x₁, y₁), the other point (x₂, y₂). After that, I set up the distance equation and replace both y(s) with their respective x(s) as follow:

After that I got stuck ... since I don't know how to differentiate an equation with two variables. Here I need to find the minimum of d, which is the distance between the two points.
I wonder if my approach to the problem is wrong or what. Any suggestion is highly appreciated. Thank you.

And just a small question: I see lots of people type equations in our forum directly. Could you please show me how to do this? Every time I need to type a math formula, I have to open Word and it took me lots of time :( Thanks again.

 

[ideasrule]

To minimize with respect to 2 variables, you just take the partial derivative of d with respect to x1, set it to 0, then take the partial derivative with respect to x2 and set that to 0. However, it's easier to minimize d^2, because you don't have to deal with the square root.

And just a small question: I see lots of people type equations in our forum directly. Could you please show me how to do this? Every time I need to type a math formula, I have to open Word and it took me lots of time :( Thanks again.

PF supports LaTex. See this for a tutorial: http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/

With simple expressions, it's usually easier to write them without LaTex.

 

[smashX]

Thank you very much for your clear suggestion. Actually, I don't know what in the world is partial differentiation (I'm just in my first semester of calculus ^^), so I just googled it and I get it now. Basically, I think it looks just like implicit differentiation. Anyway, after doing as you instructed, I got to this point:

(x₁² + (x₂ - 4)²) (x₂ - 4 + x₁) = 0

Solving the first part of the equation, I got x₁ = 0 and x₂ = 4. I usually trust myself but in this case, it seems a little bit weird. Does that mean the line segment between the two vertices (actually it's on the x-axis too!) is the shortest one? If you had some time, would you please take a look at it? Somehow I doubt this result ... sorry for the trouble and thank you once again.

And by the way, is there any other way to solve this problem? I doubt my instructor will get irritated if I turn in something that is solved by using the second-semester material ... he told us to do this by using what we learned in the section Optimization Problem of the Applications of Differentiation chapter.

 

[mege]

Try a test case:

I picked x₁ = 1 resulting in (1,1)₁ and x₂ = 4 resulting in (4,0)₂. The distance between those two points is 10^1/2 (squareroot of 10). That is less than 4 using (0,0)₍₁₎ to (4,0)₍₂₎ which are the points you found.

 

[smashX]

I see, so my results were wrong after all hmm
I tried to differentiate again but I didn't find any errors.
Any other suggestions please, thank you.