2.10 Q-2

Question Statement

Divide 20 into two parts such that the sum of their squares will be minimized.

---

Background and Explanation

This problem asks us to divide a number (in this case, 20) into two parts and minimize the sum of the squares of these parts. To solve this, we will use the concept of optimization and calculus. Specifically, we’ll define a function for the sum of squares, differentiate it, find the critical points, and use the second derivative test to confirm the minimum.

---

Solution

1. Define the variables:

   Let the two parts be $x$ and $20 - x$, where $x$ is one of the parts, and $20 - x$ is the other part.

2. Set up the sum of squares function:

   The sum of the squares of the two parts can be written as:

$$f(x) = x^2 + (20 - x)^2$$

This equation represents the sum of squares of the two parts.

3. Simplify the function:

   Expanding the squares:

$$f(x) = x^2 + (20 - x)^2 = x^2 + (400 - 40x + x^2)$$

Combining like terms:

$$f(x) = 2x^2 - 40x + 400$$

4. Find the derivative of the function:

   To minimize the function, we differentiate $f(x)$ with respect to $x$:

$$f'(x) = 4x - 40$$

5. Set the derivative equal to zero:

   To find the critical points, set the derivative equal to zero:

$$4x - 40 = 0$$

6. Solve for $x$:

   Solving the equation:

$$4x = 40 \quad \Rightarrow \quad x = 10$$

7. Verify the minimum:

   To confirm that this value of $x$ gives a minimum, check the second derivative:

$$f''(x) = 4$$

Since $f''(x)$ is positive, this indicates that $x = 10$ corresponds to a minimum.

8. Find the two parts:

   The two parts are $x = 10$ and $20 - 10 = 10$.

Thus, the required two parts of 20 are 10 and 10, which minimize the sum of their squares.

---

Key Formulas or Methods Used

• Sum of squares function: $f(x) = x^2 + (20 - x)^2$
• Derivative: $f'(x) = 4x - 40$
• Second derivative test: $f''(x) = 4$

---

Summary of Steps

1. Define the parts as $x$ and $20 - x$.
2. Write the sum of squares function: $f(x) = 2x^2 - 40x + 400$.
3. Differentiate to get $f'(x) = 4x - 40$.
4. Set the derivative equal to zero: $4x - 40 = 0$, solving gives $x = 10$.
5. Check the second derivative to confirm a minimum.
6. The parts are 10 and 10, minimizing the sum of their squares.