If n is even and `nge4,x_(1),x_(2),...,x_(n)ge0` and `x_(1)+x_(2)+...+x_(n)=1,` then `P=x_(1)x_(2)x_(3)…x_n` cannot exceed

Options

A. `(1)/(n+1)`
B. `(1)/(n+2)`
C. `(1)/(2n)`
D. none of these

Answer

To solve the problem step by step, we will use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step 1: Understand the Given Information
We are given that:
- \( n \) is even and \( n \geq 4 \)
- \( x_1, x_2, \ldots, x_n \geq 0 \)
- \( x_1 + x_2 + \ldots + x_n = 1 \)

### Step 2: Apply the AM-GM Inequality
According to the AM-GM inequality, for any non-negative numbers \( x_1, x_2, \ldots, x_n \), we have:
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
This implies:
\[ \frac{1}{n} \geq \sqrt[n]{P} \]
where \( P = x_1 x_2 \ldots x_n \).

### Step 3: Rearranging the Inequality
From the inequality above, we can rearrange it to express \( P \):
\[ P \leq \left(\frac{1}{n}\right)^n \]

### Step 4: Conclusion
Thus, we conclude that the product \( P \) cannot exceed:
\[ P \leq \frac{1}{n^n} \]

### Final Answer
Therefore, the maximum value that \( P \) can achieve is \( \frac{1}{n^n} \).

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