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May 17th, 2017, 10:05 AM   #1
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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!
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May 17th, 2017, 01:30 PM   #2
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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.
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May 17th, 2017, 01:37 PM   #3
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Quote:
Originally Posted by zylo View Post
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.
I'm really sorry but i didn't understand 100%, can you please show a graphic example?
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May 17th, 2017, 04:03 PM   #4
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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.
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May 17th, 2017, 06:21 PM   #5
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Quote:
Originally Posted by zylo View Post
Sorry. I don't know how to post a sketch. I put it on paper. Then what.
1. From the text box reply window scroll down to "Additional Options".
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.
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May 18th, 2017, 06:46 AM   #6
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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.
Attached Images
File Type: jpg MMF1.JPG (45.1 KB, 12 views)
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Last edited by zylo; May 18th, 2017 at 07:44 AM.
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May 18th, 2017, 09:05 AM   #7
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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.
Attached Images
File Type: jpg MMF1BB.jpg (45.1 KB, 12 views)
File Type: jpg MMF1BBBA.jpg (64.3 KB, 6 views)
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Last edited by zylo; May 18th, 2017 at 09:28 AM.
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May 18th, 2017, 11:07 AM   #8
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Quote:
Originally Posted by zylo View Post
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.
Oh, so that is what it asked me in exercise... now i understood! Thanks a lot!
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