Answer :

Given,


The two numbers are positive.


the sum of two numbers is 16.


the sum of the squares of two numbers is minimum.


Let us consider,


x and y are the two numbers, such that x > 0 and y > 0


Sum of the numbers : x + y = 16


Sum of squares of the numbers : S = x2 + y2


Now as,


x + y = 16


y = (16-x) ------ (1)


Consider,


S = x2 + y2


By substituting (1), we have


S = x2 + (16-y)2 ------ (2)


For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.


Differentiating the equation (2) with x




----- (3)


[Since ]


Now equating the first derivative to zero will give the critical point c.


So,



2x – 2(16 -x) = 0


2x – 32 + 2x = 0


= 4x = 32



x = 8


As x > 0, x = 8


Now, for checking if the value of S is maximum or minimum at x=8, we will perform the second differentiation and check the value of at the critical value x = 8.


Performing the second differentiation on the equation (3) with respect to x.





[Since and ]



Now when x = 8,



As second differential is positive, hence the critical point x = 8 will be the minimum point of the function S.


Therefore, the function S = sum of the squares of the two numbers is minimum at x = 8.


From Equation (1), if x= 8


y = 16 – 8 = 8


Therefore, x = 8 and y = 8 are the two positive numbers whose su is 16 and the sum of the squares is minimum.


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