In `a DeltaABC, cos""(A)/(2)cos""(B)/(2)cos""C/(2)` is

Options

A. `le(3sqrt3)/(16)`
B. `le(3sqrt3)/(8)`
C. `le(3sqrt3)/(32)`
D. `(3sqrt3)/(64)`

Explanation

To solve the problem, we need to determine the value of the expression \( \frac{\cos(A/2) \cos(B/2) \cos(C/2)}{1} \) in triangle \( ABC \).

### Step-by-Step Solution:

1. **Understanding the Property of Cosines in a Triangle:**
In any triangle \( ABC \), the sum of the cosines of the angles is given by the inequality:
\[ \cos A + \cos B + \cos C \leq \frac{3}{2} \]

2. **Adding 1 to Both Sides:**
If we add 1 to both sides of the inequality, we have:
\[ 1 + \cos A + 1 + \cos B + 1 + \cos C \leq \frac{3}{2} + 3 \]
Simplifying this gives:
\[ 3 + \cos A + \cos B + \cos C \leq \frac{9}{2} \]

3. **Expressing in Terms of Half-Angle Cosines:**
We can express \( 1 + \cos A \) as:
\[ 1 + \cos A = 2 \cos^2\left(\frac{A}{2}\right) \]
Similarly, for \( B \) and \( C \):
\[ 1 + \cos B = 2 \cos^2\left(\frac{B}{2}\right) \]
\[ 1 + \cos C = 2 \cos^2\left(\frac{C}{2}\right) \]

4. **Substituting Back into the Inequality:**
Substituting these into our inequality gives:
\[ 2 \cos^2\left(\frac{A}{2}\right) + 2 \cos^2\left(\frac{B}{2}\right) + 2 \cos^2\left(\frac{C}{2}\right) \leq \frac{9}{2} \]
Dividing the entire inequality by 2:
\[ \cos^2\left(\frac{A}{2}\right) + \cos^2\left(\frac{B}{2}\right) + \cos^2\left(\frac{C}{2}\right) \leq \frac{9}{4} \]

5. **Applying the AM-GM Inequality:**
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\cos^2\left(\frac{A}{2}\right) + \cos^2\left(\frac{B}{2}\right) + \cos^2\left(\frac{C}{2}\right)}{3} \geq \sqrt[3]{\cos^2\left(\frac{A}{2}\right) \cos^2\left(\frac{B}{2}\right) \cos^2\left(\frac{C}{2}\right)} \]
Thus, we have:
\[ \frac{9}{4} \geq 3 \sqrt[3]{\cos^2\left(\frac{A}{2}\right) \cos^2\left(\frac{B}{2}\right) \cos^2\left(\frac{C}{2}\right)} \]

6. **Finding the Upper Bound:**
Rearranging gives:
\[ \cos^2\left(\frac{A}{2}\right) \cos^2\left(\frac{B}{2}\right) \cos^2\left(\frac{C}{2}\right) \leq \left(\frac{3}{4}\right)^3 = \frac{27}{64} \]
Therefore, taking the square root:
\[ \cos\left(\frac{A}{2}\right) \cos\left(\frac{B}{2}\right) \cos\left(\frac{C}{2}\right) \leq \frac{3\sqrt{3}}{8} \]

### Conclusion:
Thus, the expression \( \cos\left(\frac{A}{2}\right) \cos\left(\frac{B}{2}\right) \cos\left(\frac{C}{2}\right) \) is bounded above by \( \frac{3\sqrt{3}}{8} \).