If `x + y + z =a` and the minimum value of `a/x+a/y+a/z` is `81^lambda`, then the value of . `lambda` is

Options

A. `(1)/(2)`
B. 1
C. `(1)/(4)`
D. 2

Answer

To solve the problem, we start with the given equation and analyze the expression we need to minimize.

### Step 1: Understand the given equation
We have:
\[ x + y + z = a \]
We need to find the minimum value of:
\[ \frac{a}{x} + \frac{a}{y} + \frac{a}{z} \]

### Step 2: Rewrite the expression
We can factor out \( a \) from the expression:
\[ \frac{a}{x} + \frac{a}{y} + \frac{a}{z} = a \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \]

### Step 3: Substitute \( a \)
Since \( a = x + y + z \), we can substitute this into our expression:
\[ a \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) = (x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \]

### Step 4: Apply the Cauchy-Schwarz inequality
Using the Cauchy-Schwarz inequality:
\[ (x+y+z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \geq (1 + 1 + 1)^2 = 9 \]
This tells us that:
\[ \frac{a}{x} + \frac{a}{y} + \frac{a}{z} \geq 9 \]

### Step 5: Find the minimum value
The minimum value of \( \frac{a}{x} + \frac{a}{y} + \frac{a}{z} \) occurs when \( x = y = z \). In this case, if \( x = y = z = k \), then \( a = 3k \) and:
\[ \frac{a}{x} + \frac{a}{y} + \frac{a}{z} = 3 \cdot \frac{3k}{k} = 9 \]

### Step 6: Set the minimum value equal to \( 81^\lambda \)
According to the problem, the minimum value is given as:
\[ 9 = 81^\lambda \]
We can rewrite \( 81 \) as \( 3^4 \):
\[ 9 = (3^4)^\lambda = 3^{4\lambda} \]
Since \( 9 = 3^2 \), we have:
\[ 3^2 = 3^{4\lambda} \]

### Step 7: Equate the exponents
This gives us:
\[ 2 = 4\lambda \]
Solving for \( \lambda \):
\[ \lambda = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = \frac{1}{2} \]