Divide 16 into two parts such that the sum of their cubes is minimum.

Answer

To solve the problem of dividing 16 into two parts such that the sum of their cubes is minimized, we can follow these steps:

### Step 1: Define the Variables
Let the two parts be \( x \) and \( y \). According to the problem, we have:
\[ x + y = 16 \]

### Step 2: Express One Variable in Terms of the Other
From the equation \( x + y = 16 \), we can express \( y \) in terms of \( x \):
\[ y = 16 - x \]

### Step 3: Write the Function to Minimize
We need to minimize the sum of the cubes of these two parts:
\[ S = x^3 + y^3 \]
Substituting the expression for \( y \):
\[ S = x^3 + (16 - x)^3 \]

### Step 4: Expand the Cubic Expression
Now, we will expand \( (16 - x)^3 \):
\[ (16 - x)^3 = 16^3 - 3 \cdot 16^2 \cdot x + 3 \cdot 16 \cdot x^2 - x^3 \]
Calculating \( 16^3 \):
\[ 16^3 = 4096 \]
Calculating \( 3 \cdot 16^2 \):
\[ 3 \cdot 16^2 = 3 \cdot 256 = 768 \]
Thus, we have:
\[ S = x^3 + (4096 - 768x + 48x^2 - x^3) \]
This simplifies to:
\[ S = 4096 - 768x + 48x^2 \]

### Step 5: Differentiate the Function
To find the minimum, we differentiate \( S \) with respect to \( x \):
\[ \frac{dS}{dx} = -768 + 96x \]

### Step 6: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ -768 + 96x = 0 \]
Solving for \( x \):
\[ 96x = 768 \implies x = \frac{768}{96} = 8 \]

### Step 7: Find the Value of \( y \)
Using \( x = 8 \) in the equation \( y = 16 - x \):
\[ y = 16 - 8 = 8 \]

### Step 8: Verify Minimum with Second Derivative Test
Now, we check if this critical point is a minimum by finding the second derivative:
\[ \frac{d^2S}{dx^2} = 96 \]
Since \( \frac{d^2S}{dx^2} > 0 \), this indicates that \( S \) has a minimum at \( x = 8 \).

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

### Summary
The two parts into which 16 can be divided such that the sum of their cubes is minimized are both 8.

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