Question

10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectively

• A. 9, 1
• B. 8, 2
• C. 6, 4
• D. 7, 3

Hint:

When an optimization question has simple integer choices, you can bypass calculus entirely and check the options directly!

• Option (A) [9 and 1]: $2(9) + 1^2 = 18 + 1 = 19$

• Option (B) [8 and 2]: $2(8) + 2^2 = 16 + 4 = 20$

• Option (D) [7 and 3]: $2(7) + 3^2 = 14 + 9 = 23$
Since 19 is the lowest value among the calculated sums, option (A) is instantly verified as the minimum.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The number 10 is partitioned into two distinct values. We need to find these two components such that the mathematical function modeling "twice the first part plus the square of the second part" achieves its absolute minimum value.

Step 2: Key Formula or Approach:
1. Let the first part be $y$ and the second part be $x$. Since their sum is 10, we have $y = 10 - x$.
2. Write down the objective function $f(x)$ representing the conditions: $$f(x) = 2y + x^2 = 2(10 - x) + x^2$$ 3. To find the minimum point, find the first derivative $f'(x)$, set it to zero to get the critical values, and verify using the second derivative test ($f''(x) > 0$).

Step 3: Detailed Explanation:
4. Set up the explicit quadratic function of $x$: $$f(x) = 20 - 2x + x^2 = x^2 - 2x + 20$$ 5. Differentiate the function with respect to $x$: $$f'(x) = 2x - 2$$ 6. Set the first derivative to zero to locate the extremum position: $$2x - 2 = 0 \implies 2x = 2 \implies x = 1$$ 7. Perform the second derivative test to confirm it is a local minimum: $$f''(x) = \frac{d}{dx}(2x - 2) = 2$$ Since $2 > 0$ for all values, the function forms an upward-opening parabola, confirming that a strict minimum occurs at $x = 1$. 8. Calculate the corresponding first part $y$: $$y = 10 - x = 10 - 1 = 9$$ Thus, the two numerical parts are 9 and 1 respectively.

Step 4: Final Answer:
The two numbers are 9 and 1 respectively, matching option (A).