There are many pairs of numbers (positive and negative) of which the sum is worth the unit. Of those, find the 2 numbers whose sum, double the square of the first number and the square of the 2nd number would give a minimal value.
To solve this problem, let's break it down into steps:
Step 1: Define the two numbers
Let's assume the two numbers are x and y. Since they can be both positive and negative, we can express them as +/- x and +/- y, respectively.
Step 2: Set up the equation
According to the problem, the sum of these two numbers is equal to 1. So we can write the equation as:
(+/- x) + (+/- y) = 1
Step 3: Define the expression to minimize
We need to minimize the expression: 2x^2 + y^2. Let's call it F(x, y).
Step 4: Substitute the values
Replace x and y in the expression F(x, y) with (+/- x) and (+/- y) respectively:
F(x, y) = 2(+/- x)^2 + (+/- y)^2
Step 5: Simplify the expression
F(x, y) = 2x^2 + y^2
Step 6: Use calculus to find the minimum
To find the minimum of the expression F(x, y), we need to take partial derivatives with respect to x and y and set them equal to zero:
dF/dx = 4x = 0
dF/dy = 2y = 0
From the first equation, we get x = 0, and from the second equation, we get y = 0.
Step 7: Find the values of x and y
Since x = 0 and y = 0 minimize the expression, the two numbers (-x, -y) that give the minimal value are (-0, -0), which is just (0, 0).
Therefore, the two numbers whose sum, double the square of the first number, and the square of the second number would give a minimal value are 0 and 0.