SOLUTION: The sum of two numbers is 18. If the sum of their squares is a minimum find the numbers.

Question 132033: The sum of two numbers is 18. If the sum of their squares is a minimum find the numbers.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
, or put another way, .

The sum of the two numbers squares is , but because of the first equation, can be expressed as , which is a function in x.

Expand the binomial:

Two ways to find a minimum for this function. The algebra way is to recognize that this is a convex up parabola and solve for the vertex. The calculus way is to set the first derivitive equal to zero and solve.

The x-coordinate of the vertex of a parabola in the form is found by evaluating .

, so the sum of squares is a minimum when x = 9 => y = 9.

Calculus method:
Take the first derivitive
Set it equal to zero and solve

, so f has a local extreme point at x = 9

Now take the second derivitive
, which is positive, therefore the local extreme is a minimum. So the function is minimum when x = 9 => y = 9.

Same answer.
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