Rules - Maxima and Minima

Rule 1: For positive variables, if the sum of the variables is a constant, the product of the variables will be maximum when all the variables are equal. 

Eg: If a + b + c = 21, find the maximum value of abc.

Here sum of the variable is constant.  So product will be maximum, when all the three variables are equal 

i.e., 3a = 21, a = 7.  So product = 7 x 7 x 7 = 343

Rule 2: For positive variables, if the product of the variables is a constant, the sum of the variables will be minimum when all the variables are equal

Eg: Find the minimum value of ab+bc+ca${\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}}$

Here the product of the variables = ab×bc×ca${\displaystyle \frac{a}{b}\times \frac{b}{c}\times \frac{c}{a}}$ = 1

So given sum is minimum when all are equal ab=1,bc=1,ca=1${\displaystyle \frac{a}{b}=1,\frac{b}{c}=1,\frac{c}{a}=1}$

So sum = 1 + 1 + 1 = 3. 
ab+bc+ca${\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}}$≥$\ge$3

Rule 3: For positive variables, Arithmetic Mean (AM), is always greater than Geometric Mean (GM) i.e., A.M≥G.M$A.M\ge G.M$

Eg: If xy = 16, then find the minimum value of x + y. 
AM of x, y = x+y2${\displaystyle \frac{x+y}{2}}$
GM of x, y = (x.y)−−−−√$\sqrt{\left(x.y\right)}$
from AM GM rule x+y2≥(x.y)−−−−√${\displaystyle \frac{x+y}{2}\ge \sqrt{\left(x.y\right)}}$
Substituting xy = 16, we get x+y2≥4${\displaystyle \frac{x+y}{2}\ge 4}$
Or x+y≥8