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Solution for If X+Y=8, Then the Maximum Value of Xy is (A) 8 (B) 16 (C) 20 (D) 24 - CBSE (Science) Class 12 - Mathematics

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Question

If x+y=8, then the maximum value of xy is
(a) 8
(b) 16
(c) 20
(d) 24

Solution

\[(b)  16\] 
\[\text { Given }:   x + y = 8\] 
\[ \Rightarrow y = 8 - x                                                       .  .  .   \left( 1 \right)\] 
\[\text { Let }  f\left( x \right) \text { be }  xy . \] 
\[ \Rightarrow   f\left( x \right) =   x\left( 8 - x \right)                          \left[ \text { From  eq } . \left( 1 \right) \right]\] 
\[ \Rightarrow f'\left( x \right) = 8 - 2x\] 
\[\text { For  a  local  maxima  or  a  local  minima,   we  must  have }  \] 
\[f'\left( x \right) = 0\] 
\[ \Rightarrow 8 - 2x = 0\] 
\[ \Rightarrow 8 = 2x\] 
\[ \Rightarrow x = 4\] 
\[ \Rightarrow y = 8 - 4 = 4                          \left[ \text { From  eq } . \left( 1 \right) \right]\] 
\[\text { Now,} \] 
\[f''\left( x \right) =  - 2\] 
\[ \Rightarrow f''\left( 4 \right) =  - 2 < 0\] 
\[\text { So,   x = 4  is  a  local  maxima }. \] 
\[\text { Hence,   the  local  maximumvalue  is  given  by }\] 
\[f\left( 4 \right) = 4 \times 4 = 16\]
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Solution If X+Y=8, Then the Maximum Value of Xy is (A) 8 (B) 16 (C) 20 (D) 24 Concept: Graph of Maxima and Minima.
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