• 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
x + y = 16
y = (16-x) ------ (1)
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
Now equating the first derivative to zero will give the critical point c.
⇒ 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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