Advanced High-School Mathematics

SECTION 3.2 Classical Inequalities 147

-4 -2 2 4

-4

-2

2

4

x +y =c

xy=2

xx

22

x

As a final variation on the above y
theme, note can that can interchange
the roles of constraint and “objective
function” and ask for the extreme val-
ues of x^2 +y^2 given the constraint
xy= 2. The relevant figure is given
to the right. Notice that there is no
maximum ofx^2 +y^2 , but that the min-
imum value is clearly x^2 +y^2 = 4,
again occurring at the points of tan-
gency.

Exercises.

1. Find the maximum of the functionxygiven the elliptical constraint
   4 x^2 +y^2 = 6. Draw the constraint graph and the “level curves”
   whose equations arexy=constant.


2. Given that xy = −5, find the maximum value of the objective
   functionx^2 + 3y^2.


3. Given that xy = 10, find the maximum value of the objective
   functionx+y.


4. Suppose thatxandyare positive numbers with
   x+y= 1. Compute the minimum value of

(

1 +

1

x

) (

1 +

1

y

)

3.2 Classical Unconditional Inequalities

Until further notice, we shall assume that the quantitiesx 1 , x 2 , ..., xn
are all positive. Define

Arithmetic Mean:

AM(x 1 ,x 2 ,...,xn) =

x 1 +x 2 +···+xn

n

;

Geometric Mean:

GM(x 1 ,x 2 ,...,xn) = n

√

x 1 x 2 ···xn;