How to find the minimum value of an equation?

[annemsujan]

For example how to find the minimum value of the expression 2x+y in which x*y=72 ?
In detail for what values of x and y the expression gives minimum value?

[Math Tutor]

[tex]y = 72/x[/tex]

[tex]f(x) = 2x + 72/x[/tex]

we have to find the min. of the function

[tex]f'(x) = 2 - \frac{72}{ x^2} = 0[/tex]


[tex]x = \pm 6[/tex]

f increase when x < -6 in -6 it has a local maximum
then decrease in the interval (-6; +6) and it has a local minimum in +6
then increase in x > 6

So the min is x=6

[martosss]

from AM-GM we obtain [tex]2x+y\ge2\sqrt{2xy}=2*\sqrt{144}=2*12=24[/tex]. Equality when 2x=y=12.

[broniran]

from AM-GM we obtain [tex]2x+y\ge2\sqrt{2xy}=2*\sqrt{144}=2*12=24[/tex]. Equality when 2x=y=6.

But [tex]x,y[/tex] don't have to be positive ;]

[MM]

Let [tex]2x+y=a[/tex]. Then we have to find the minimal [tex]a[/tex] for which [tex]\left(a-2x\right)x=72[/tex]. Examining the last as quadratic equation towards [tex]x[/tex] we get that its discriminant is [tex]D=a^{2}-576[/tex] which must be nonnegative since [tex]x[/tex] is real. From the inequality [tex]a^{2}-576\ge 0[/tex] we get [tex]a\in\left(-\infty;-24\right)\cup\left(24;+\infty\right)[/tex]. So [tex]2x+y[/tex] doesn't reach minimal value. For example take [tex]x=-8[/tex] and [tex]y=-9[/tex]. Then [tex]2x+y=-25[/tex].

[martosss]

from AM-GM we obtain [tex]2x+y\ge2\sqrt{2xy}=2*\sqrt{144}=2*12=24[/tex]. Equality when 2x=y=6.

But [tex]x,y[/tex] don't have to be positive ;]

ou, yes, they do!
If [tex]x,y<0[/tex], then there isn't a min at all, because you can have
a=-10 00...0
b=-72/10 00...0
Now you have ab=72, but 2a+b=-20 00...0 ... an so on. Do you get the picture?

You can have a-> oo, b-> 0 and still ab=72.
That is why we need a,b>0

[broniran]

The things you are saying are bullshits... There isn't any condition for x and y being positive. If you think that minimum doesn't simply write that it doesn't but not assume things, which hadn't been given in the problem...

[poster123]

But x,y don't have to be positive ;]

[Guest]

increase when x < -6 in -6 it has a local maximum
then decrease in the interval (-6; +6) and it has a local minimum in +6
then increase in x > 6

[sajid121]

There isn't any condition for x and y being positive. If you think that minimum doesn't simply write that it doesn't but not assume things, which hadn't been given in the problem..

[sajid121]

this group and don't want to start off on a negative note, but I have been assisting for a long time in several other newsgroups, and am here to tell you that "kate" sends her questions to more than one forum, and never show any personal effort. I like to distinguish between help and do-my-homework