Primarygun Posted August 18, 2006 Share Posted August 18, 2006 Given, [Math]S=x^{3}y^{2}z[/Math] and [Math]x+y+z=1[/Math] Determine the greatest value of S. How do we start this type of question? I started with [Math]S=xxxyyz[/Math]where there are six terms; used AM-GM and found that x=y=z, I know it's wrong, but why? Then I think it occurs because there's no limits of them. Therefore, I sub [Math]z=1-x-y[/Math] So [Math]S=(1-x-y)(x^{3}y^{2})[/Math] But there's no clear solution for me. What should I do next? Link to comment Share on other sites More sharing options...

The Thing Posted August 23, 2006 Share Posted August 23, 2006 Let y = a negative number with a huge absolute value. Link to comment Share on other sites More sharing options...

Dave Posted August 23, 2006 Share Posted August 23, 2006 This isn't an inequalities concept. And the above post isn't really appropriate, since we're looking for a specific value of S. What you want to be looking at is the method of Lagrange multipliers, which allows you to find extrema of a function f(x,y,z) subject to a constraint g(x,y,z) = 0. In this case, [math]f(x,y,z) = S = x^3y^2z[/math] and [math]g(x,y,z) = x+y+z-1[/math]. You can find out more about it by looking at the Wikipedia article. Link to comment Share on other sites More sharing options...

matt grime Posted August 23, 2006 Share Posted August 23, 2006 The above comment is appropriate: S is not bounded above, and you don't need Lagrange multipliers to see this, it is also, correspondingly, not bounded below (z=1, x=-y, and let x tend to infinity or minus infinity). Link to comment Share on other sites More sharing options...

Dave Posted August 23, 2006 Share Posted August 23, 2006 I misread the original my post. My mistake. Link to comment Share on other sites More sharing options...

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