Show that of all the rectangles with a given perimeter, the square has the largest area.
Let us Consider a rectangle with length is x cm and breadth is y cm.
Since, Area of rectangle = length × breadth
And, Perimeter of rectangle is , = 2(length + breadth)
P=2(x + y) …(ii)
Now, From (ii),
Putting the value of y in equation (ii), we get
Now, differentiate A, w.r.t x , we get
Corresponds to a maximum value of A.
The area is maximum when the sides are
Hence, This is the square, with largest area.
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