In `!ABC` , least value of `(e^(A))/A+(e^(B))/B+(e^(C))/C` is equal to

Options

A. `9/pie^(pi//3)`
B. `pi/3e^(pi//3)`
C. `pi/9e^(pi//3)`
D. none of these

Explanation

To find the least value of the expression \(\frac{e^A}{A} + \frac{e^B}{B} + \frac{e^C}{C}\) in triangle \(ABC\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here are the steps to derive the solution:

### Step-by-Step Solution:

1. **Understanding the Expression**: We need to minimize the expression \(\frac{e^A}{A} + \frac{e^B}{B} + \frac{e^C}{C}\).

2. **Applying AM-GM Inequality**: According to the AM-GM inequality, for any positive numbers \(x_1, x_2, x_3\):
\[
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