Smallest possible value of the sum of their squares is 128.
Maximum and Minimum value:
The minimum is the smallest value in the data set,while maximum is the largest value in the data set.
Let,
x and y be the required numbers.
and S be the sum of squares of these values,
According to question,
x + y = 16 ----------- (I)
and,
[tex]S =x^2 +y^2[/tex] -------------(II)
from (I) and (II)
S = [tex]x^2 + (16 - x)^2[/tex]
By Simplifying the above equation using algebraic identity
[tex](a-b)^2 = a^2+ b^2 -2ab[/tex] , we get
[tex]S = x^2 + 16^2 +x^2 -2*16*x\\\\ = x^2 + 256 +x^2 -32x\\\\= 2x^2 -32x +256[/tex]
Thus,
[tex]S= 2x^2 -32x +256\\\\[/tex]
[tex]S = 2(x^2 -16x +128)\\\\ = 2[ x^2 -2*8*x +8^2 +64]\\ \\ = 2[ (x-8)^2 + 64]\\\\S = 2[ (x-8)^2 + 64][/tex]
Value of S will be minimum when, x=8
at x=8
S = 2[0 + 64]
= 2 *64
=128
Smallest value of S is 128
Thus,
Smallest possible value of the sum of their squares is 128.
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