Find maximum value of a function in two variables

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September 1st 2011, 11:18 AM
shridhar18

Find maximum value of a function in two variables

Hi,

I have a problem.

Given a+b=10.
Find maximum value of a^3*b^2

How do i go about doing this?

Also i heard,

a^x*b^y*c^z is maximum when a/x=b/y=c/z, provided a+b+c=constant?

Can someone give me the proof? Also i am appearing for an exam that tests me in such applications of math, can someone post a link to the most common axioms and proofs?

Thanks.
September 1st 2011, 12:52 PM
TKHunny

Re: Find maximum value of a function in two variables

There are various ways to go about it. The simplest might be a quick substitution. a = 10-b might lead somewhere useful.
September 1st 2011, 09:54 PM
shridhar18

Re: Find maximum value of a function in two variables

Quote:

Originally Posted by TKHunny

There are various ways to go about it. The simples might be a quick substitution. a = 10-b might lead somewhere useful.

The substituion is a valid process. But i wanted to know if i could apply the concept i described earlier. The reason is this problem needs to be solved in 1 minute. If the concept(theorem/axiom) described earlier is accurate, could anybody explain to me the proof or a high level reasoning.

Thanks
September 1st 2011, 09:56 PM
shridhar18

Re: Find maximum value of a function in two variables

By the concept i meant --> The max value of a^x.b^y is obtained when a/x=b/y.
September 2nd 2011, 03:47 AM
TKHunny

Re: Find maximum value of a function in two variables

Ummm...That may be right, but I'm not sure it's useful. It has very limited scope (solves very few problems) and is unlikely to appear in that form ever again - unless you already know what will be on the exam.
September 2nd 2011, 04:48 AM
shridhar18

Re: Find maximum value of a function in two variables

True about the scope. But can you help me deduce or prove the theorem. I would want to know how they arrived at the concept. That helps me understand the direction in which to approach the problem. And some variations of the problem can be expected.

Also any links to pages explaining some useful theorems? WHat i am looking for is a site that just lists down some salient and useful theorems. Eg

1+2+3+...+n = [n(n+1)]/2

Thanks.
September 2nd 2011, 02:44 PM
TKHunny

Re: Find maximum value of a function in two variables

How is your experience with Lagrangian Multipliers? If it can be proven, that should work.
September 3rd 2011, 12:52 AM
shridhar18

Re: Find maximum value of a function in two variables

Quote:

Originally Posted by TKHunny

How is your experience with Lagrangian Multipliers? If it can be proven, that should work.

I can't recollect that. Are you suggesting Lagrange's to deduce the theorem or to solve the problem? Either way could you explain the same?

THanks
September 3rd 2011, 04:00 AM
HallsofIvy

Re: Find maximum value of a function in two variables

To maximize $f(a,b,c)= a^xb^yc^z$ subject to the condition that g(a, b, c)= a+ b+ c= constant, make the gradients parallel:
$\nabla f= xa^{x-1}b^yc^z\vec{i}+ ya^xb^{y-1}c^z\vec{j}+ za^xb^yz^{c-1}\vec{k}= \lambda\nabla g= \lambda\vec{i}+ \lambda\vec{j}+ \lambda\vec{k}$
where $\lambda$ is the Lagrange multiplier.

Then we must have $xa^{x-1}b^y c^z= \lambda$ , $ya^xb^{y-1}c^z= \lambda$ , and $za^xb^yc^{z-1}= \lambda$ .

We can eliminate $\lambda$ from those equations by dividing one by another: Dividing the first equation by the second, $\frac{xa^{x-1}b^yc^z}{ya^xb^{y-1}c^z}= \frac{xb}{ya}= 1$ or $\frac{x}{a}= \frac{y}{b}$ . Dividing the second equation by the third gives $\frac{ya^xb^{y-1}c^z}{za^xb^yc^{z-1}}= 1$ or $\frac{y}{b}= \frac{z}{c}$ .

If you are unfamiliar with "Lagrange Multipliers", here's a way to think about them- if you want to maximize f(x,y,z), calculate $\nabla f$ , which always points in the direction of fastest increase of f, and follow it until you can't follow it any more- until it is 0. If you are constrained to stay on surface g(x,y,z)= constant, you may not be able to move in that direction ( $\nabla f$ may point off the surface), but you could move in the direction of the projection of $\nabla f$ onto the surface. You can do that, getting higher values of f, until there is NO projection on the surface- that is, until $\nabla f$ is perpendicular to the surface. But that means that it will be parallel to the normal vector to g at that point: $\nabla f$ is parallel to $\nabla g$ or $\nabla f= \lambda\nabla g$ for some number $\lambda$ .
September 3rd 2011, 08:49 AM
TKHunny

Re: Find maximum value of a function in two variables

That's why I asked about your experience with it before demonstrating it. :-)
September 3rd 2011, 12:03 PM
shridhar18

Re: Find maximum value of a function in two variables

Quote:

Originally Posted by TKHunny

That's why I asked about your experience with it before demonstrating it. :-)

I think you just spoke Greek to me. :-). But going by the comments at first glance, you basically are asking to find the point at which the tangent to the curve(3D in this case) is parallel to the axis right?

Or have i just completely missed the point?