Create an image illustrating a mathematical concept. Depict a graph on an x-y plane representing the function of two variables: one on the x-axis and the other on the y-axis. Show that the sum of these two variables totals 10, with a unique point marked that indicates the minimum sum of their squares. Add markers, lines, and colors to highlight this key point and make the graph visually appealing, but remember to include no text.

Find two numbers whose sum is 10 for which the sum of their squares is a minimum.

x and (10-x)

s = x^2 + (10-x)^2
s = x^2 + 100 -20x + x^2
s = 2 x^2 - 20 x + 100
s/2 = x^2 -10 x + 100 we can minimize half the sum easier than the whole sum
That is a parabola and you could find the vertex but since you said this was "calculus" we will take the derivative and set to zero.
0 = 2 x - 10
x = 5
10-x = 5

two numbers : x and 10-x

S = x^2 + (10-x)^2
dS/dx = 2x - 2(10-x)
= 0 for max/min
x = 5

so the numbers are both 5

Interesting the answer is halfway between.

Exploring that
Say a sum of two numbers is s
We want to minimize the sum of squares of x^2 and (s-x)^2
sum = 2 x^2 -2sx
d sum/dx = 0 = 4 x -2s
x = s/2
so it works for any old sum, not just 10

Well, if you're looking for a minimum sum of squares, you'll have to delve into the fascinating world of complex numbers!

Let me introduce you to our two main characters, "x" and "y." They're our number buddies who, when combined, give us the magical sum of 10. We have the equation x + y = 10.

Now, let's bring in their squares: x^2 and y^2. In our quest for a minimum sum of squares, we want to minimize the individual squares. So, we need to minimize x^2 and y^2.

Since we can't give preference to either x or y, let's assume x and y are equal. So, x = y.

Substituting that into our original equation, we get 2x = 10, which means x = 5 and y = 5.

Thus, the two numbers whose sum is 10 and the sum of their squares is a minimum are 5 and 5.

Remember, even in the world of complex numbers, sometimes simplicity wins!

To find two numbers whose sum is 10 for which the sum of their squares is a minimum, we can use calculus.

Let's suppose the two numbers are x and 10 - x, where x represents one of the numbers. We want to minimize the sum of their squares, which can be expressed as x^2 + (10 - x)^2.

To find the minimum value, we can take the derivative of this expression with respect to x and set it equal to zero. Then, solve for x.

Differentiating the expression, we get:
d/dx (x^2 + (10 - x)^2) = 2x - 2(10 - x)

Setting the derivative equal to zero:
2x - 2(10 - x) = 0

Simplifying the equation:
2x - 20 + 2x = 0
4x - 20 = 0
4x = 20
x = 5

Now we have the value of x as 5. Substituting this back into the expression for the two numbers, we find that the two numbers are 5 and 5.

Thus, the two numbers whose sum is 10 for which the sum of their squares is a minimum are 5 and 5.