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.
