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 apply the Cauchy-Schwarz inequality. Here, we will use the Cauchy-Schwarz inequality for a more straightforward approach.

### Step 1: Apply the Cauchy-Schwarz Inequality

According to the Cauchy-Schwarz inequality, we have:

\[ \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)(a + b + c) \geq (1 + 1 + 1)^2 \]

### Step 2: Substitute the Given Condition

Since we know that \( a + b + c = 1 \), we can substitute this into the inequality:

\[ \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)(1) \geq 3^2 \]

### Step 3: Simplify the Inequality

This simplifies to:

\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]

### Step 4: Determine When Equality Holds

The equality in Cauchy-Schwarz holds when \( \frac{1}{a} = \frac{1}{b} = \frac{1}{c} \), which implies that \( a = b = c \). Given that \( a + b + c = 1 \), we can set:

\[ a = b = c = \frac{1}{3} \]

### Step 5: Calculate the Minimum Value

Now substituting \( a = b = c = \frac{1}{3} \) into the expression \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \):

\[ \frac{1}{\frac{1}{3}} + \frac{1}{\frac{1}{3}} + \frac{1}{\frac{1}{3}} = 3 + 3 + 3 = 9 \]

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

### Final Answer

The least value of \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) is \( \boxed{9} \).

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