The sum of two positive numbers is 10. Find these numbers if the sum of
their squares is a maximum.
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
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
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.
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.
sir, you are correct.
But I think you ascribe too much mathematical sophistication to the humble student.
To find the two positive numbers whose sum is 10 and the sum of their squares is a maximum, we can use the method of optimization.
Let's assume the two positive numbers as x and 10 - x (since their sum is 10).
Now, we need to maximize the sum of their squares, which is given by the equation: S = x^2 + (10 - x)^2.
To find the maximum value of S, we can take the derivative of S with respect to x and set it equal to zero.
dS/dx = 2x - 2(10 - x) = 0
Simplifying the equation:
2x - 20 + 2x = 0
4x - 20 = 0
4x = 20
x = 5
Now we have found the value of x = 5.
Using this value, we can find the other number:
10 - x = 10 - 5 = 5
So, the two positive numbers whose sum is 10 and the sum of their squares is a maximum are 5 and 5.