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?

Let y = a negative number with a huge absolute value.

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.

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).

I misread the original my post. My mistake.