If a, b, c are positive and a + b + c = 1, then find the least value of `(1)/(a)+(2)/(b)+(3)/(c)`

Options

A. 9
B. 18
C. 27
D. 6

Answer

To find the least value of the expression \(\frac{1}{a} + \frac{2}{b} + \frac{3}{c}\) given that \(a + b + c = 1\) and \(a, b, c\) are positive, we can follow these steps:

### Step 1: Understand the constraints
We know that \(a\), \(b\), and \(c\) are positive numbers and their sum is equal to 1. This means \(a, b, c > 0\) and \(a + b + c = 1\).

**Hint:** Remember that since \(a\), \(b\), and \(c\) are positive, they cannot be zero or negative.

### Step 2: Use the method of Lagrange multipliers or Cauchy-Schwarz inequality
We can apply the Cauchy-Schwarz inequality to minimize the expression. According to the Cauchy-Schwarz inequality:

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

### Step 3: Substitute the constraint
Since \(a + b + c = 1\), we substitute this into the inequality:

\[ \left( \frac{1}{a} + \frac{2}{b} + \frac{3}{c} \right)(1) \geq 6 \]

This simplifies to:

\[ \frac{1}{a} + \frac{2}{b} + \frac{3}{c} \geq 6 \]

**Hint:** The equality holds when the ratios are equal, which means \(\frac{1}{a} : \frac{2}{b} : \frac{3}{c} = 1 : 2 : 3\).

### Step 4: Set the ratios
From the ratios, we can set:

\[ \frac{1}{a} = k, \quad \frac{2}{b} = 2k, \quad \frac{3}{c} = 3k \]

This gives us:

\[ a = \frac{1}{k}, \quad b = \frac{2}{2k} = \frac{1}{k}, \quad c = \frac{3}{3k} = \frac{1}{k} \]

### Step 5: Find a, b, and c
Now, we can express \(a\), \(b\), and \(c\) in terms of \(k\):

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

From this, we find \(k = 3\). Therefore:

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

### Step 6: Substitute back into the expression
Now substitute \(a\), \(b\), and \(c\) back into the expression:

\[ \frac{1}{a} + \frac{2}{b} + \frac{3}{c} = \frac{1}{\frac{1}{3}} + \frac{2}{\frac{1}{3}} + \frac{3}{\frac{1}{3}} = 3 + 6 + 9 = 18 \]

### Conclusion
Thus, the least value of \(\frac{1}{a} + \frac{2}{b} + \frac{3}{c}\) is \(18\).

**Final Answer:** 18