Question 1190186
.
The sum of two non-negative numbers is 30. Find the numbers if:
(a) The sum of their squares is as large as possible; as small as possible
(b) The square of one number plus the square root of the other number is as large as possible; as small as possible
~~~~~~~~~~~~~~~~



            I will answer/solve question/problem  (a),  ONLY.



<pre>
Let one number be 15+x;  then the other number is 15-x.



The sum of their squares is then

    (15+x)^2 + (15-x)^2 = (225 + 2x + x^2) + (225 - 2x + x^2) = 450 + 2x^2.



The addend 2x^2 is always non-negative and is minimal at x= 0.

The sum of the squares is as small as possible at x= 0, when the sum of the squares is 450.

This minimum is provided when both the numbers are equal to 15.



The maximum of the sum of squares is reached at x = +/-15, when x^2 is as large as possible.

Under this condition, the numbers itself are 0 and 30.
</pre>

Parts (a) is solved.