Optimization

Step 1: Set up the variables.
Let the two parts into which the number 20 is divided be and . The sum of the squares of these parts is given by:

Step 2: Simplify the expression.
Expanding the expression:

Step 3: Minimize the function.
To minimize , we take the derivative with respect to and set it equal to zero: Setting the derivative equal to zero:

Step 4: Verify that this is a minimum.
Taking the second derivative: Since the second derivative is positive, the function has a minimum at .

Step 5: Conclude.
Thus, the number 20 should be divided into two parts of 10 and 10 to minimize the sum of their squares.

Final Answer: