Find the Minimum Value of: (18)/(a+b) + (12)/(ab) + 8a + 5b.where a and b are positive real numbers I tried evaluating the expression into 1 denominator, and tried to get squares so i can evaluate to 0 but obviously it was the wrong approach.

metal1fc 2022-09-13 Answered
Minimum value in 2 variables
Find the Minimum Value of:
18 a + b + 12 a b + 8 a + 5 b .
where a and b are positive real numbers
I tried evaluating the expression into 1 denominator, and tried to get squares so i can evaluate to 0 but obviously it was the wrong approach.
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Answers (1)

Makhi Adams
Answered 2022-09-14 Author has 13 answers
By AM_GM we obtain:
18 a + b + 12 a b + 8 a + 5 b = ( 18 a + b + 2 ( a + b ) ) + ( 12 a b + 6 a + 3 b )
2 18 a + b 2 ( a + b ) + 3 12 a b 6 a 3 b 3 = 30.
The equality occurs for ( a , b ) = ( 1 , 2 ), which says that 30 is a minimal value.

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