Maximize the value of the function z = <mfrac> <mrow> a b +

Bronson Olson

Bronson Olson

Answered question

2022-05-03

Maximize the value of the function
z = a b + c a + b + c ,
where a , b , c are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, and if possible a general technique. Thank you.

Answer & Explanation

Kendrick Fritz

Kendrick Fritz

Beginner2022-05-04Added 12 answers

z = a b + c a + b + c = a b a b a + b + c + 1
Clearly, this will be maximum if c minimum =1

So,
z a b a b a + b + 1 + 1 = a b + 1 a + b + 1
a 1 b 1 + 1 a 1 + b 1 + 1 will be greater than a 2 b 2 + 1 a 2 + b 2 + 1
if ( a 1 a 2 1 ) ( b 1 b 2 ) + ( b 1 b 2 1 ) ( a 1 a 2 ) + a 1 b 1 a 2 b 2 > 0
if ( a 1 a 2 1 ) ( b 1 b 2 ) + ( b 1 b 2 1 ) ( a 1 a 2 ) + ( a 1 a 2 ) b 1 + a 2 ( b 1 b 2 ) > 0
if ( a 1 a 2 1 + a 2 ) ( b 1 b 2 ) + ( b 1 b 2 1 + b 1 ) ( a 1 a 2 ) > 0
Clearly, a b + 1 a + b + 1 increases with the increment of a , b or both.

So, a b + 1 a + b + 1 will be maximum if a , b are maximum.

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