October 2nd, 2017, 09:27 AM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 178 Thanks: 2  Getting all maxima and mimina...
The equation 2x^24x+y^2 = 16 describes an ellipse in the plane. (a) Find the point(s) on this ellipse closest to the origin. Answer (2, 0). (b) Find the point(s) on this ellipse farthest from the origin. Answer: (2, 4) and (2, 4). $\displaystyle f(x,y,\lambda) = x^2 + y^2  \lambda ( 2x^2  4x + y^2  16 ) $ The above equation yields only (2, 4) and (2, 4), but it does not yields (2, 0) at all.. Would anyone be able to show me how to get (2, 0) with lagrange multipler? 
October 2nd, 2017, 10:31 AM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,038 Thanks: 423 
Try the correct equation $L(x,\ y,\ \lambda ) = \sqrt{x^2 + y^2}  \lambda (2x^2  4x + y^2  16).$ 
October 2nd, 2017, 10:38 AM  #3 
Senior Member Joined: Jan 2017 From: Toronto Posts: 178 Thanks: 2  
October 2nd, 2017, 07:09 PM  #4 
Senior Member Joined: May 2016 From: USA Posts: 1,038 Thanks: 423 
First off, technically, you cannot represent an ellipse by a single function. That may or may not be relevant, but you cannot be sure whether it is relevant unless you take it into account. Second, the functions representing an ellipse have bounded domains. Again that may or may not be relevant, but you cannot be sure unless you take it into account. Third, you do not need to use Lagrangian multipliers for this problem. The ellipse is described by: $2x^2  4x + y^2 = 16.$ So the real functions that are relevant are: $a(x) = y = +\ \sqrt{16 + 4x  2x^2} \text { such that } \ 2 \le x \le 4 \text { and}$ $b(x) = y = \ \sqrt{16 + 4x  2x^2} \text { such that } \ 2 \le x \le 4.$ Why those bounds? $\pm \sqrt{16 + 4x  2x^2} = y \in \mathbb R \implies 16 + 4x  2x^2 \ge 0 \implies 16 \ge 2x^2  4x \implies$ $8 \ge x^2  2x \implies 9 \ge x^2  2x + 1 \implies 3^2 \ge (x  1)^2 \implies$ $3 \ge (x  1) \ge \ 3 \implies 4 \ge x \ge \ 2.$ The distance function between the origin and a(x) is: $c(x) = \sqrt{(x  0)^2 + \left ( +\ \sqrt{16 + 4x  2x^2}  0 \right )^2} = \sqrt{ 16 + 4x  x^2 }.$ $c'(x) = \dfrac{4  2x}{2\sqrt{ 16 + 4x  x^2 }} \implies c'(x) = 0 \iff x = 2.$ $x = 2 \implies y = \sqrt{16 + 4(2)  2(2)^2} = \sqrt{16 + 8  8} = 4.$ $distance = c(2) = \sqrt{ 16 + 4(2)  (2)^2 } = \sqrt{16 + 8  4} = 2\sqrt{5}.$ So we may have a local extremum at (2, 4). The distance function between the origin and b(x) is: $d(x) = \sqrt{(x  0)^2 + \left ( \ \sqrt{16 + 4x  2x^2}  0 \right )^2} = \sqrt{ 16 + 4x  x^2 }.$ $d'(x) = \dfrac{4  2x}{2\sqrt{ 16 + 4x  x^2 }} \implies d'(x) = 0 \iff x = 2.$ $x = 2 \implies y = \ \sqrt{16 + 4(2)  2(2)^2} = \sqrt{16 + 8  8} = \ 4.$ $distance = d(2) = \sqrt{ 16 + 4(2)  (2)^2 } = \sqrt{16 + 8  4} = 2\sqrt{5}.$ So may have a local extremum at (2,  4).. So, the distance may be at an extremum at both (2, 4) and (2,  4). This would be possible only if the locus was a perfect circle centered on the origin, but it is not. What is wrong? WITH A BOUNDED FUNCTION, WE MUST ALSO TESTS THE BOUNDS. So what is c(4)? $c(4) = \sqrt{16 + 4(4)  (4)^2} = \sqrt{16 + 16  16} = 4 < 2\sqrt{5}.$ And what is c( 2)? $c(\ 2) = \sqrt{16 + 4(\ 2)  (\ 2)^2} = \sqrt{16  8  4} = \sqrt{4} = 2 < 4 < 2\sqrt{5}.$ $b( 2) = \sqrt{16 + 4(2)  2(\ 2)^2} = \sqrt{16  8  8} = 0.$ So we have maximal distance at (2, 4) and (2,  4) and minimal distance at ( 2, 0). Derivatives are not enough with bounded functions. Last edited by JeffM1; October 2nd, 2017 at 07:15 PM. 

Tags 
maxima, mimina 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Minima and Maxima  Volle  Calculus  3  July 28th, 2015 01:12 PM 
MinimaMaxima  Volle  Calculus  4  July 21st, 2015 12:48 AM 
Integrating in Maxima  PhizKid  Math Software  1  December 26th, 2012 05:11 AM 
Maxima  Dontlookback  Calculus  2  May 20th, 2010 12:37 AM 
Maxima or minima  ArmiAldi  Real Analysis  1  March 6th, 2008 04:51 AM 