Find the maximum value of the product of two real numbers

AI Thread Summary
The maximum value of the product of two real numbers, given their sum is 12, is determined to be 36 using the arithmetic and geometric means inequality. When solving for the conditions where x equals y, it is confirmed that both values are 6, leading to the maximum product. Alternative methods, including direct substitution and the Lagrange multipliers technique, also yield the same result, reinforcing the conclusion that the maximum occurs when the numbers are equal. The discussion emphasizes the symmetrical nature of the equations involved, suggesting that equal values correspond to extrema. Overall, the maximum product of 36 is consistently validated through various mathematical approaches.
chwala
Gold Member
Messages
2,825
Reaction score
413
Homework Statement
The sum of two real numbers ##x## and ##y## is ##12##. Find the maximum value of their product ##xy##.
Relevant Equations
Arithmetic and geometric means
Using the inequality of arithmetic and geometric means,
$$\frac {x+y}{2}≥\sqrt{xy}$$
$$6^2≥xy$$
$$36≥xy$$

I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.
 
Physics news on Phys.org
Solve
xy=36
x+y=12
to check your assumption.
 
Last edited:
anuttarasammyak said:
Solve
xy=36
x+y=12
to check your assumption.
$$x^2-12x+36=0$$, Giving us a repeated root... i get that, ok ##⇒x=y## cheers Anutta...
 
Another way to do this that doesn't use the arithmetic mean and geometric mean:

Maximize ##f(x, y) = xy## given that ##x + y = 12##.
##f(x, y) = xy = x(12 - x) = -x^2 + 12x = -(x^2 - 12x + 36) + 36 = -(x - 6)^2 + 36##
The graph of the last expression is a parabola that opens downward, with its vertex at (6, 36). Since the parabola opens downward, its maximum value is 36.
 
  • Like
  • Love
Likes benorin, Delta2 and chwala
Yet another way is direct substitution: use ##x+y =6## to sub in in ##f(x,y)= xy## and get a function of x alone. Then find the max in the usual way.
 
  • Like
Likes benorin, Delta2 and chwala
WWGD said:
Yet another way is direct substitution: use ##x+y =6## to sub in in ##f(x,y)= xy## and get a function of x alone. Then find the max in the usual way.
This is exactly what I did in my previous post.
 
  • Like
  • Love
Likes Delta2 and WWGD
In both ##x+y=12## and ##xy##, ##x## and ##y## appear symmetrically, so I'd expect ##x=y## to correspond to an extremum ##xy##. It's then easy to see that it's a maximum and not a minimum.
 
  • Like
Likes benorin and Delta2
vela said:
In both ##x+y=12## and ##xy##, ##x## and ##y## appear symmetrically, so I'd expect ##x=y## to correspond to an extremum ##xy##. It's then easy to see that it's a maximum and not a minimum.
Good point. Guess we can model it by a rectangle with perimeter 24 ( simplified by halving), whose area is maximized when both its sides are equal.
 
  • Like
Likes benorin
...I just noted that we could also use the Lagrange Multiplier, in solving this kind of problems...
 
  • Like
Likes benorin and PhDeezNutz
  • #10
Lagrange multipliers method:

##f(x,y)=xy##
##g(x,y)=x+y=12##

##\nabla f := \lambda \nabla g##
##<y,x> := \lambda <1,1>##
Hence ##x=\lambda =y\Rightarrow x+x=12\Rightarrow x=6=y##

Not so bad :)
 
  • Like
Likes chwala
  • #11
benorin said:
Lagrange multipliers method:

##f(x,y)=xy##
##g(x,y)=x+y=12##

##\nabla f := \lambda \nabla g##
##<y,x> := \lambda <1,1>##
Hence ##x=\lambda =y\Rightarrow x+x=12\Rightarrow x=6=y##

Not so bad :)
Nice,...##g(x,y)= x+y##...from what I've read... and not ##x+y-12##...Is that correct. Cheers man!
 
  • #12
the derivative of a constant is zero so it doesn't matter
 
  • #13
benorin said:
the derivative of a constant is zero so it doesn't matter
Thanks ...I had already noted that...I just want to be certain on the correct definition of ##g(x,y)##...cheers benorin.
 
  • #14
Don't think of it as a function ##g(x,y)## think of it as a restriction ##g(x,y):=k## is an equation not a function, more precisely it's level curve or surface of the function I told you not to think of. ;)
 
  • Like
Likes chwala
Back
Top