If sum of two numbers is 3, then maximum value of the product of first and the square of second is

Options

A. 4
B. 3
C. 2
D. 1

Explanation

To solve the problem of finding the maximum value of the product of the first number and the square of the second number, given that their sum is 3, we can follow these steps:

### Step 1: Define the Variables
Let the first number be \( x \) and the second number be \( y \). According to the problem, we have:
\[ x + y = 3 \]
From this equation, we can express \( y \) in terms of \( x \):
\[ y = 3 - x \]

### Step 2: Define the Product Function
We need to maximize the product \( P \) defined as:
\[ P = x \cdot y^2 \]
Substituting \( y \) from Step 1 into the product function, we get:
\[ P = x \cdot (3 - x)^2 \]

### Step 3: Expand the Product Function
Now, let's expand the product:
\[ P = x \cdot (9 - 6x + x^2) = 9x - 6x^2 + x^3 \]

### Step 4: Differentiate the Product Function
To find the maximum value, we differentiate \( P \) with respect to \( x \):
\[ \frac{dP}{dx} = 9 - 12x + 3x^2 \]

### Step 5: Set the Derivative to Zero
To find the critical points, we set the derivative equal to zero:
\[ 3x^2 - 12x + 9 = 0 \]
Dividing the entire equation by 3 simplifies it to:
\[ x^2 - 4x + 3 = 0 \]

### Step 6: Solve the Quadratic Equation
Now, we can factor the quadratic:
\[ (x - 3)(x - 1) = 0 \]
This gives us the solutions:
\[ x = 3 \quad \text{or} \quad x = 1 \]

### Step 7: Determine the Corresponding Values of \( y \)
Using \( y = 3 - x \):
- If \( x = 3 \), then \( y = 0 \).
- If \( x = 1 \), then \( y = 2 \).

### Step 8: Calculate the Product for Each Case
Now we can calculate \( P \) for both values of \( x \):
1. For \( x = 3 \) and \( y = 0 \):
\[ P = 3 \cdot 0^2 = 0 \]
2. For \( x = 1 \) and \( y = 2 \):
\[ P = 1 \cdot 2^2 = 1 \cdot 4 = 4 \]

### Step 9: Conclusion
The maximum value of the product \( P \) occurs when \( x = 1 \) and \( y = 2 \), resulting in:
\[ \text{Maximum value of } P = 4 \]