Optimization by Equalization

Idea · Intermediate

Algebra

a,b\ge 0

Fixed quantity	Quantity optimized	Minimum	Occurs when	Maximum	Occurs when
$a+b=S$	$ab$	$0$	$a,b$ are as far apart as possible	$\dfrac{S^2}{4}$	$a,b$ are as close as possible
$a+b=S$	$a^2+b^2$	$\dfrac{S^2}{2}$	$a,b$ are as close as possible	$S^2$	$a,b$ are as far apart as possible
$ab=P>0$	$a+b$	$2\sqrt P$	$a,b$ are as close as possible	No maximum	$a,b$ become arbitrarily far apart
$ab=P>0$	$a^2+b^2$	$2P$	$a,b$ are as close as possible	No maximum	$a,b$ become arbitrarily far apart
$a^2+b^2=Q$	$a+b$	$\sqrt Q$	$a,b$ are as far apart as possible	$\sqrt{2Q}$	$a,b$ are as close as possible
$a^2+b^2=Q$	$ab$	$0$	$a,b$ are as far apart as possible	$\dfrac Q2$	$a,b$ are as close as possible

Key Idea

For optimization problems involving two nonnegative numbers:

\boxed{\text{Look at what happens when }a\text{ and }b\text{ become closer together or farther apart}}

If a and b are real numbers, as close as possible often means

a=b

If a and b are integers, equality may be impossible. In that case, choose them as close as possible.

For example, if

a+b=15,\qquad a,b\in\mathbb Z_{\ge0}

the closest possible pair is

\boxed{7,8}

not 7.5, 7.5.

For n positive integer and nonnegative real variables a₁, a₂, ..., aₙ, the following generalizations hold. When the product is fixed at P > 0, all variables are positive.

Fixed quantity	Quantity optimized	Minimum	Occurs when	Maximum	Occurs when
$\displaystyle\sum_{i=1}^{n}a_i=S$	$\displaystyle\prod_{i=1}^{n}a_i$	$0$	Variables are as far apart as possible	$\left(\dfrac Sn\right)^n$	Variables are as close as possible
$\displaystyle\sum_{i=1}^{n}a_i=S$	$\displaystyle\sum_{i=1}^{n}a_i^2$	$\dfrac{S^2}{n}$	Variables are as close as possible	$S^2$	Variables are as far apart as possible
$\displaystyle\prod_{i=1}^{n}a_i=P>0$	$\displaystyle\sum_{i=1}^{n}a_i$	$nP^{1/n}$	Variables are as close as possible	No maximum	Variables become arbitrarily far apart
$\displaystyle\prod_{i=1}^{n}a_i=P>0$	$\displaystyle\sum_{i=1}^{n}a_i^2$	$nP^{2/n}$	Variables are as close as possible	No maximum	Variables become arbitrarily far apart
$\displaystyle\sum_{i=1}^{n}a_i^2=Q$	$\displaystyle\sum_{i=1}^{n}a_i$	$\sqrt Q$	Variables are as far apart as possible	$\sqrt{nQ}$	Variables are as close as possible
$\displaystyle\sum_{i=1}^{n}a_i^2=Q$	$\displaystyle\prod_{i=1}^{n}a_i$	$0$	Variables are as far apart as possible	$\left(\dfrac Qn\right)^{n/2}$	Variables are as close as possible