If `a_(1),a_(2),a_(3)` be any positive real numbers, then which of the following statement is not true.

Options

A. `3a_(1),a_(2),a_(3)lea_(1)^(3)+a_(2)^(3)+a_(3)^(3)`
B. `(a_(1))/(a_(2))+(a_(2))/(a_(3))+(a_(3))/(a_(1))ge3`
C. `(a_(1)+a_(2)+a_(3))((1)/(a_(1))+(1)/(a_(2))+(1)/(a_(3)))ge9`
D. `(a_(1)+a_(2)+a_(3))((1)/(a_(1))+(1)/(a_(2))+(1)/(a_(3)))^(3)le27`

Answer

To determine which statement is not true among the given options regarding the positive real numbers \( a_1, a_2, a_3 \), we can utilize the properties of inequalities, specifically the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here’s a step-by-step solution:

### Step 1: Understand the AM-GM Inequality
The AM-GM inequality states that for any non-negative real numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
with equality if and only if all \( x_i \) are equal.

### Step 2: Apply AM-GM to \( a_1, a_2, a_3 \)
For the positive real numbers \( a_1, a_2, a_3 \):
\[ \frac{a_1 + a_2 + a_3}{3} \geq \sqrt[3]{a_1 a_2 a_3} \]
This implies:
\[ a_1 + a_2 + a_3 \geq 3\sqrt[3]{a_1 a_2 a_3} \]

### Step 3: Apply AM-GM to the Reciprocals
Now, consider the reciprocals \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3} \):
\[ \frac{\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}}{3} \geq \sqrt[3]{\frac{1}{a_1 a_2 a_3}} \]
This simplifies to:
\[ \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} \geq \frac{3}{\sqrt[3]{a_1 a_2 a_3}} \]

### Step 4: Multiply the Two Inequalities
Now, we multiply the two inequalities obtained:
\[ (a_1 + a_2 + a_3) \left(\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}\right) \geq 9 \]
This is derived from:
\[ \left(3\sqrt[3]{a_1 a_2 a_3}\right) \left(\frac{3}{\sqrt[3]{a_1 a_2 a_3}}\right) = 9 \]

### Step 5: Identify the Incorrect Statement
From the above derivation, we conclude that:
\[ (a_1 + a_2 + a_3) \left(\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}\right) \geq 9 \]
Thus, any statement contradicting this conclusion is not true.

### Conclusion
We can analyze the given options to find which one does not hold true based on our derived inequality.