Find two number whose sum is 100 and the sum of twice of first part and square of the second is minimum.

Answer

To solve the problem of finding two numbers whose sum is 100, and where the sum of twice the first number and the square of the second number is minimized, we can follow these steps:

### Step 1: Define the variables
Let the first number be \( x \) and the second number be \( y \).

### Step 2: Set up the equation for the sum
According to the problem, the sum of the two numbers is:
\[ x + y = 100 \]
From this, we can express \( y \) in terms of \( x \):
\[ y = 100 - x \]

### Step 3: Set up the function to minimize
We need to minimize the function:
\[ S = 2x + y^2 \]
Substituting the expression for \( y \):
\[ S = 2x + (100 - x)^2 \]

### Step 4: Expand the function
Now, we expand the function:
\[ S = 2x + (10000 - 200x + x^2) \]
Combining like terms, we get:
\[ S = x^2 - 198x + 10000 \]

### Step 5: Differentiate the function
To find the minimum, we differentiate \( S \) with respect to \( x \):
\[ \frac{dS}{dx} = 2x - 198 \]

### Step 6: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 2x - 198 = 0 \]
Solving for \( x \):
\[ 2x = 198 \quad \Rightarrow \quad x = 99 \]

### Step 7: Find the value of \( y \)
Now, substitute \( x = 99 \) back into the equation for \( y \):
\[ y = 100 - x = 100 - 99 = 1 \]

### Step 8: Conclusion
Thus, the two numbers are:
\[ x = 99 \quad \text{and} \quad y = 1 \]

### Summary
The two numbers whose sum is 100 and minimizes the sum of twice the first number and the square of the second number are \( 99 \) and \( 1 \).

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