If ` a + b + c = 9 ` (where a, b, c, are real numbers) , then the minimum value of ` a^(2) + b^(2) + c^(2)` is

Options

A. 100
B. 9
C. 27
D. 81

Answer

To find the minimum value of \( a^2 + b^2 + c^2 \) given that \( a + b + c = 9 \), we can use the method of Lagrange multipliers or the Cauchy-Schwarz inequality. However, a simpler approach is to use the concept of equality among the variables.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to minimize the expression \( a^2 + b^2 + c^2 \) under the constraint \( a + b + c = 9 \).

2. **Using the Method of Equal Variables**: To minimize the sum of squares, we can assume that \( a \), \( b \), and \( c \) are equal. This is because for a fixed sum, the sum of squares is minimized when the numbers are as close to each other as possible.

3. **Setting the Variables Equal**: Let \( a = b = c = x \). Then, from the constraint \( a + b + c = 9 \), we have:
\[ 3x = 9 \]

4. **Solving for x**: Dividing both sides by 3 gives:
\[ x = 3 \]

5. **Substituting Back to Find the Minimum Value**: Now substituting \( x \) back into the expression for the sum of squares:
\[ a^2 + b^2 + c^2 = x^2 + x^2 + x^2 = 3x^2 \]
Substituting \( x = 3 \):
\[ 3(3^2) = 3 \times 9 = 27 \]

6. **Conclusion**: Therefore, the minimum value of \( a^2 + b^2 + c^2 \) is \( 27 \).

### Final Answer:
The minimum value of \( a^2 + b^2 + c^2 \) is \( 27 \).