May 17th, 2017, 11:05 AM  #1 
Member Joined: Mar 2017 From: Israel Posts: 30 Thanks: 2  Tangent
Hello Can you help me please about the following exercise: There is an elevation f(x, y)=ln(x)+ln(y) when x, y > 0. Show that solving an extreme problem: max{ln(x)+ln(y)} s.t. x+2y=24 sits on point of tangency. The intention is to present only a graphical solution. Thanks a lot! 
May 17th, 2017, 02:30 PM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,215 Thanks: 91 
The given line projected vertically intersects the surface in a curve. A tangent to the curve is tangent to the surface. At an extremum of the curve the tangent is horizontal and thus parallel to the given line. 
May 17th, 2017, 02:37 PM  #3 
Member Joined: Mar 2017 From: Israel Posts: 30 Thanks: 2  I'm really sorry but i didn't understand 100%, can you please show a graphic example?

May 17th, 2017, 05:03 PM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,215 Thanks: 91 
Sorry. I don't know how to post a sketch. I put it on paper. Then what. But anyhow: Draw x,y,z axes. Draw a spherical dome above x,y plane. Draw a line in x,y plane below dome. Draw a vertical plane through the line. It intersects the dome in a curve (circle). Sketch tangents along the curve. At the high point of the curve (circle) the tangent is horizantal and in the same direction as the original line. 
May 17th, 2017, 07:21 PM  #5  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,697 Thanks: 977 Math Focus: Elementary mathematics and beyond  Quote:
2. Click on the "Manage Attachments" button under "Attach Files". 3. Under "Upload File from your Computer" click on the "Choose File" button. 4. Select the file you wish to attach. 5. Click on the "Upload" button. That completes the attachment process.  
May 18th, 2017, 07:46 AM  #6 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,215 Thanks: 91 
1 and 2 are extrema on surface along given line. Sorry, don't know how to rotate file. Rotated it in jpeg original but didn't help. Thanks greg1313 for instructions. Last edited by zylo; May 18th, 2017 at 08:44 AM. 
May 18th, 2017, 10:05 AM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,215 Thanks: 91 
TaDah. Open file on desktop. Rotate. Save copy with new name. Upload. EDIT Second image expands larger. Open original in jpeg window. Choose "open" "paint." In paint choose "resize" and then add pixels. I added 200. Then save as jpeg image. Last edited by zylo; May 18th, 2017 at 10:28 AM. 
May 18th, 2017, 12:07 PM  #8 
Member Joined: Mar 2017 From: Israel Posts: 30 Thanks: 2  Oh, so that is what it asked me in exercise... now i understood! Thanks a lot!


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