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 their cubes, which can be expressed as:
\[ S = x^3 + y^3 \]
Substituting \( y \) from Step 2, we get:
\[ S = x^3 + (16 - x)^3 \]

### Step 4: Expand the Function
Now, we expand \( (16 - x)^3 \):
\[ (16 - x)^3 = 16^3 - 3 \times 16^2 \times x + 3 \times 16 \times x^2 - x^3 \]
Calculating \( 16^3 \):
\[ 16^3 = 4096 \]
Calculating \( 3 \times 16^2 \):
\[ 3 \times 16^2 = 3 \times 256 = 768 \]
So, substituting back, 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 Corresponding Value of \( y \)
Using \( x = 8 \) in the equation \( y = 16 - x \):
\[ y = 16 - 8 = 8 \]

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

### Conclusion
Thus, the two parts into which 16 is divided such that the sum of their cubes is minimized are:
\[ x = 8 \quad \text{and} \quad y = 8 \]