 # 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:
$\frac{18}{a+b}+\frac{12}{ab}+8a+5b.$
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.
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By AM_GM we obtain:
$\frac{18}{a+b}+\frac{12}{ab}+8a+5b=\left(\frac{18}{a+b}+2\left(a+b\right)\right)+\left(\frac{12}{ab}+6a+3b\right)\ge$
$\ge 2\sqrt{\frac{18}{a+b}\cdot 2\left(a+b\right)}+3\sqrt{\frac{12}{ab}\cdot 6a\cdot 3b}=30.$
The equality occurs for $\left(a,b\right)=\left(1,2\right)$, which says that $30$ is a minimal value.

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