Question

The sum of two positive numbers is 10. Find these numbers if the sum of
their squares is a maximum.

Answer 1

a + b = 10
y = a^2 + b^2
y = a^2 + (10-a)^2
y = 2 a^2 - 20 a +100 = 2 (a^2 -10 a + 50)
dy/da = 0 at max or min = 2 [ 2a - 10 }
so
a = 5
b = 5

Answer 2

Point ( 5 , 5 ) is. minimum of function

y = 2 a² 20 a +100

You also must doo second derivative test.

If y" < 0 then function has a local maximum.

If y" > 0 then function has a local minimum.

In this case y" = 4 > 0

So for a = 5 , b = 5 function

y = 2 a² 20 a +100

has a minimum.

You can check that.

a = 1 , b = 9 , a² + b² = 82

a = 2 , b = 8 , a² + b² = 68

a = 3 , b = 7 , a² + b² = 58

a = 4 , b = 6 , a² + b² = 52

a = 5 , b = 5 , a² + b² = 50

a = 6 , b = 4 is same as

a = 4 , b = 5 , a² + b² = 52

a = 7 , b = 3 is same as

a = 7 , b = 3 , a² + b² = 58

etc.

For a = 5 , b = 5

y = 2 a² 20 a +100 has a minimum.

The function y = 2 a² 20 a +100

has no maximum

Answer 3

surely there is a maximum, as both numbers cannot exceed 10!
0^2 + 10^2 = 100
1^2 + 9^2 = 82
...
5^2 + 5^2 = 50
So clearly 1,9 gives the maximum sum of squares.

Answer 4

Assuming the numbers he is looking for are whole numbers, but it didn't
say that.
So, we have a poorly worded question.

I can find a pair that have a higher max than 1,0
how about .005, 9.995 , for a sum of squares of 99.90005

the sum of their squares will approach 100, but never get there, since
one of the numbers being zero cannot happen.

Answer 5

sir, you are correct.
But I think you ascribe too much mathematical sophistication to the humble student.