Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are

Options

A. `(10,10)`
B. `(5,15)`
C. `(13,7)`
D. none of these

Explanation

To solve the problem of dividing 20 into two parts such that the product of one part and the cube of the other is maximized, we can follow these steps:

### Step 1: Define the Variables
Let the two parts be \( x \) and \( y \). According to the problem, we have:
\[ x + y = 20 \]

### Step 2: Express the Function to Maximize
We need to maximize the function \( f(x, y) = x \cdot y^3 \). Since we have a relationship between \( x \) and \( y \), we can express \( x \) in terms of \( y \):
\[ x = 20 - y \]
Now substitute this into the function:
\[ f(y) = (20 - y) \cdot y^3 \]

### Step 3: Simplify the Function
Expanding the function gives:
\[ f(y) = 20y^3 - y^4 \]

### Step 4: Differentiate the Function
To find the maximum, we need to differentiate \( f(y) \) with respect to \( y \):
\[ f'(y) = \frac{d}{dy}(20y^3 - y^4) = 60y^2 - 4y^3 \]

### Step 5: Set the Derivative to Zero
To find the critical points, set the derivative equal to zero:
\[ 60y^2 - 4y^3 = 0 \]
Factoring out common terms:
\[ 4y^2(15 - y) = 0 \]
This gives us:
\[ y^2 = 0 \quad \text{or} \quad 15 - y = 0 \]
Thus, the critical points are:
\[ y = 0 \quad \text{or} \quad y = 15 \]

### Step 6: Determine the Nature of the Critical Points
To determine whether these points are maxima or minima, we can use the second derivative test. First, we find the second derivative:
\[ f''(y) = \frac{d}{dy}(60y^2 - 4y^3) = 120y - 12y^2 \]

Now, evaluate the second derivative at the critical points:
1. For \( y = 0 \):
\[ f''(0) = 120(0) - 12(0)^2 = 0 \] (Inconclusive)

2. For \( y = 15 \):
\[ f''(15) = 120(15) - 12(15)^2 = 1800 - 2700 = -900 \] (Negative, indicating a maximum)

### Step 7: Find the Corresponding Value of the Other Part
Since \( y = 15 \), we can find \( x \):
\[ x = 20 - y = 20 - 15 = 5 \]

### Conclusion
The two parts are:
\[ x = 5 \quad \text{and} \quad y = 15 \]

### Final Answer
The two parts are 5 and 15.

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