Results 1 to 3 of 3

Thread: Point on the graph minimize problem

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    9

    Point on the graph minimize problem

    Consider the function below.
    Find the point on the graph of function that is closest to the point (-3, 12).
    ( , 2)

    I know you have to use the distance formula and plug in your equation in for the y value. Differentiate simplify and set it equal to zero to minimize and solve for x. I keep getting stuck on the differentiation and solving for x.


    thanks in advance for any help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    Here's a very useful hint. The distance formula has a square root in it, as you know, and thus is frustrating to differentiate. However take note of the fact that if you minimize $\displaystyle D^2$, you also minimize D. So try that out and if you're still stuck show your work and we'll find your error.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, thebacontalks2me!

    Consider the function: .$\displaystyle f(x) \:=\:(x+9)^2 + 9$

    Find the point on the graph of $\displaystyle f(x)$ that is closest to the point $\displaystyle A(\text{-}3, 12)$
    Here's another useful procedure . . .

    Let the point be $\displaystyle P(x,y)$, where $\displaystyle y \:=\:(x+9)^2 + 9 \:=\:x^2 + 18x + 90$


    Using Jameson's hint, let: .$\displaystyle D \;=\;(x+3)^2 + (y - 12)^2$

    . . Differentiate: .$\displaystyle D' \;=\;2(x+3) + 2(y-12)\!\cdot\!y' \;=\;0$


    Since $\displaystyle y-12\:=\:x^2+18x+78$, and $\displaystyle y' \:=\:2x + 18$

    . . we have: .$\displaystyle 2(x + 3) + 2(x^2+18x+78)(2x+18) \;=\;0$

    . . which simplifies to: .$\displaystyle 2x^3 + 54x^2 + 481x + 1407 \:=\:0$

    . . which factors: .$\displaystyle (x+7)(2x^2 + 40x + 201) \;=\;0$

    . . and has the real root: .$\displaystyle x\:=\:\text{-}7$

    Then: .$\displaystyle y \:=\:(\text{-}7)^2 + 18(\text{-}7) + 90 \:=\:13$


    Therefore, the nearest point is: .$\displaystyle (\text{-}7,\,13)$

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: Nov 1st 2010, 05:23 AM
  2. Replies: 9
    Last Post: Jul 20th 2010, 08:00 AM
  3. Replies: 1
    Last Post: Apr 27th 2009, 02:30 PM
  4. minimize problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 9th 2008, 10:38 PM
  5. Point on a CSC graph
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: Oct 5th 2007, 08:03 AM

Search Tags


/mathhelpforum @mathhelpforum