To find the smallest value of the sum of the squares of two positive numbers whose sum is 16, let’s denote the two numbers as x and y.
We know from the problem that:
- x + y = 16
We need to minimize the expression for the sum of their squares:
- S = x2 + y2
First, we can use the constraint x + y = 16 to express y in terms of x:
- y = 16 – x
Now we can substitute this into our equation for S
- S = x2 + (16 – x)2
Expanding the equation:
- S = x2 + (256 – 32x + x2)
- S = 2x2 – 32x + 256
This is a quadratic equation in terms of x. To find the minimum value, we can use the vertex formula for a quadratic equation ax2 + bx + c, which gives the vertex (minimum point) at x = -b/(2a).
- a = 2 and b = -32
Calculating the vertex:
- x = -(-32)/(2*2) = 32/4 = 8
Now we can find y:
- y = 16 – x = 16 – 8 = 8
Thus, both numbers are equal to 8. We can now calculate the smallest sum of their squares:
- S = 82 + 82 = 64 + 64 = 128
Therefore, the smallest value of the sum of their squares is 128.