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If $xy=a_{2}$ and $S=b_{2}x+c_{2}y$ where $a,b$ and $c$ are constants, then the minimum value of $S$ is
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Text solutionVerified
we have
$xy=a_{2}$ and $S=b_{2}x+c_{2}y$
$⇒S=b_{2}x+xc_{2}a_{2} $
$⇒dxdS =b_{2}−x_{2}c_{2}a_{2} $ and $dx_{2}d_{3}S =x_{3}2c_{2}a_{2} $
For local maximum or minimum, we must have
$xdS =0⇒b_{2}−x_{2}c_{2}a_{2} =0⇒x_{2}=b_{2}c_{2}a_{2} ⇒x=±bca $
Clearly, $dx_{2}d_{2}S >0$ for $x=bca $
So, $x=bca $ is the point of local minimum
Local minimum value of $S=b_{2}(bca )+c_{2}(caa_{2}b )=2abc$
$xy=a_{2}$ and $S=b_{2}x+c_{2}y$
$⇒S=b_{2}x+xc_{2}a_{2} $
$⇒dxdS =b_{2}−x_{2}c_{2}a_{2} $ and $dx_{2}d_{3}S =x_{3}2c_{2}a_{2} $
For local maximum or minimum, we must have
$xdS =0⇒b_{2}−x_{2}c_{2}a_{2} =0⇒x_{2}=b_{2}c_{2}a_{2} ⇒x=±bca $
Clearly, $dx_{2}d_{2}S >0$ for $x=bca $
So, $x=bca $ is the point of local minimum
Local minimum value of $S=b_{2}(bca )+c_{2}(caa_{2}b )=2abc$
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Question Text  If $xy=a_{2}$ and $S=b_{2}x+c_{2}y$ where $a,b$ and $c$ are constants, then the minimum value of $S$ is

Topic  Application of Derivatives 
Subject  Mathematics 
Class  Class 12 
Answer Type  Text solution:1 
Upvotes  83 