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: Translate the words into variables.
Split 10 as second part $x$ and first part $10 - x$. We must minimise (twice the first part) plus (square of the second part): $S = 2(10 - x) + x^2$.
Step 2: Simplify the objective.
$S(x) = 20 - 2x + x^2 = x^2 - 2x + 20$.
Step 3: Complete the square instead of differentiating.
$x^2 - 2x + 20 = (x - 1)^2 + 19$. This neat form shows the structure of the minimum at a glance.
Step 4: Read off the minimum.
Since $(x-1)^2 \ge 0$ and equals 0 only when $x = 1$, the smallest value of $S$ is $19$, achieved at $x = 1$.
Step 5: Find the first part.
First part $= 10 - x = 10 - 1 = 9$.
Step 6: State the pair.
The first and second numbers are $9$ and $1$ respectively.
\[ \boxed{9 \text{ and } 1} \]