If, a,b,c are positive and a + b + c =1 , then the least value of
`1/a + 1/b +1/c ` is

Options

A. 9
B. 5
C. 3
D. 1

Answer

To find the least value of \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) given that \( a + b + c = 1 \) and \( a, b, c \) are positive, we can use the method of Lagrange multipliers or the Cauchy-Schwarz inequality. Here, we will use the Cauchy-Schwarz inequality for simplicity.

### Step-by-step Solution:

1. **Apply Cauchy-Schwarz Inequality**:
According to the Cauchy-Schwarz inequality:
\[ (x_1^2 + x_2^2 + x_3^2)(y_1^2 + y_2^2 + y_3^2) \geq (x_1y_1 + x_2y_2 + x_3y_3)^2 \]
We can set \( x_1 = \frac{1}{a}, x_2 = \frac{1}{b}, x_3 = \frac{1}{c} \) and \( y_1 = a, y_2 = b, y_3 = c \).

2. **Substituting Values**:
Thus, we have:
\[ \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)(a + b + c) \geq (1 + 1 + 1)^2 \]
Since \( a + b + c = 1 \), we can simplify this to:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]

3. **Equality Condition**:
The equality in Cauchy-Schwarz holds when \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = a : b : c \). This implies that \( a = b = c \).

4. **Finding Values of a, b, and c**:
Since \( a + b + c = 1 \) and \( a = b = c \), we can set:
\[ a = b = c = \frac{1}{3} \]

5. **Calculating the Minimum Value**:
Now substituting \( a, b, c \) back into the expression:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{\frac{1}{3}} + \frac{1}{\frac{1}{3}} + \frac{1}{\frac{1}{3}} = 3 + 3 + 3 = 9 \]

### Conclusion:
Thus, the least value of \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) is \( \boxed{9} \).