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Application Of Derivatives

Question
CBSEENMA12035510

Show that of all the rectangles with a given perimeter, the square has the largest area.

Solution

Let x , y be the lengths of sides of rectangle and 2 k be the perimeter.
therefore space space space 2 straight x plus 2 straight y space equals space 2 straight k space space space space space space space rightwards double arrow space space space space straight x plus straight y space equals space straight k
therefore space space space straight y space equals space straight k minus straight x                                                        ...(1)
Let  straight A space equals space straight x space straight y space equals space straight x left parenthesis straight k minus straight x right parenthesis space equals space space kx minus straight x squared                           open square brackets because space of space left parenthesis 1 right parenthesis close square brackets
dA over dx space equals space straight k minus 2 straight x
                 dA over dx space equals space 0 space space space give space us space straight K minus 2 straight x space minus 0 space space space space space space rightwards double arrow space space space space space straight x space equals space straight k over 2
        fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space 2 space
At space space space space space space straight x space equals space straight k over 2 space fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space 2 less than 0
therefore space space space straight A space is space maximum space when space straight x space equals space straight k over 2 comma space space straight y space equals space minus straight k minus straight k over 2 space equals space straight k over 2
∴  area of a rectangle of given perimeter is maximum when its sides are equal i.e., when it is a square.

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