Question

Prove that it is impossible to maximize the sum of two real numbers if their product is 4.​

Answer 1

Proved that it is impossible to maximize the sum of two real numbers if their product is 4.​  

Step 1:

Let say two numbers are a and b

Product = ab = 4

=> b = 4/a

Step 2:

Assume that sum is

Z = a + b

Z = a + 4/a

Need to maximize Z

Step 3:

Take derivative of Z wrt a

Z = a + 4/a

dZ/da = 1  -  4/a²

Step 4:

Equate derivate with 0

1  -  4/a² = 0

=> a² = 4

=> a = ± 2

Step 5:

Find 2nd derivative of Z

dZ/da = 1  -  4/a²

d²Z/da² =  8/a³

for a = - 2  d²Z/da²  < 0  ,

Hence Maxima at a = - 2

a  = - 2 => b = - 2

Sum = - 2 - 2 = - 4

Maxima is -4  but its local

for a =  2  d²Z/da² > 0  ,

Hence Minima at a =  2

a  =   2 => b =  2

Sum =  2 + 2 =   4

Minima is 4  

Minima is more than maxima Hence these are local Minima and Maxima.

Hence   it is impossible to maximize the sum of two real numbers if their product is 4.​

For Only positive numbers with product 4  , Minimum sum is 2 + 2 = 4

For only Negative numbers with product 4 , Maximum sum is  4

But if Both positive and negative numbers are considered then no Maxima or minima exists.

Hence , Proved that it is impossible to maximize the sum of two real numbers if their product is 4.​

Answer 2

Answer:

Suppose xy=4 with sum S. Consider the pair and 4

Clearly +4 is greater than .