If a,b,c,d,e are five positive numbers, then

Options

A. `(a/b+b/c) (c/d+d/e) ge 4 sqrt(a/e)`
B. `b/a+c/b+d/c+e/d+a/ege5`
C. `a/b+b/c+c/d+d/e+e/a lt 5`
D. None of these

Answer

To solve the problem, we need to analyze the given options based on the properties of positive numbers and apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have five positive numbers \( a, b, c, d, e \). We need to analyze four options involving these numbers and determine which of them is true.

2. **Option A**:
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} \geq 4 \sqrt{\frac{a}{e}} \]
- We can apply the AM-GM inequality to the first two terms:
\[ \frac{a}{b} + \frac{b}{c} \geq 2\sqrt{\frac{a}{c}} \]
- For the next two terms:
\[ \frac{c}{d} + \frac{d}{e} \geq 2\sqrt{\frac{c}{e}} \]
- Combining these results:
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} \geq 2\sqrt{\frac{a}{c}} + 2\sqrt{\frac{c}{e}} \]
- Now, applying AM-GM again to the two square roots:
\[ 2\sqrt{\frac{a}{c}} + 2\sqrt{\frac{c}{e}} \geq 4\sqrt[4]{\frac{a}{e}} \]
- Therefore, Option A is true.

3. **Option B**:
\[ \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{e}{d} + \frac{a}{e} \geq 5 \]
- Again, applying AM-GM:
\[ \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{e}{d} + \frac{a}{e} \geq 5\sqrt[5]{\frac{b}{a} \cdot \frac{c}{b} \cdot \frac{d}{c} \cdot \frac{e}{d} \cdot \frac{a}{e}} = 5 \]
- Thus, Option B is also true.

4. **Option C**:
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} + \frac{e}{a} < 5 \]
- We already established that the minimum value of the left-hand side is 5 (from the application of AM-GM). Therefore, Option C is false.

5. **Conclusion**:
- Options A and B are true, while Option C is false.