If a,b,c and d are four real numbers of the same sign, then the value of `(a)/(b)+(b)/(c )+(c )/(d)+(d)/(a)` lies in the interval

Options

A. `[2,oo)`
B. `[3,oo)`
C. `(4,oo)`
D. `[4,oo)`

Answer

To solve the problem, we need to find the value of the expression \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}\) given that \(a\), \(b\), \(c\), and \(d\) are four real numbers of the same sign.

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean-Geometric Mean Inequality (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 when all \(x_i\) are equal.

2. **Applying AM-GM to the Given Expression:**
We can apply the AM-GM inequality to the terms \(\frac{a}{b}\), \(\frac{b}{c}\), \(\frac{c}{d}\), and \(\frac{d}{a}\):
\[ \frac{\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}}{4} \geq \sqrt[4]{\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a}} \]

3. **Simplifying the Right Side:**
The product on the right side simplifies as follows:
\[ \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} = \frac{a \cdot b \cdot c \cdot d}{b \cdot c \cdot d \cdot a} = 1 \]
Therefore, we have:
\[ \sqrt[4]{1} = 1 \]

4. **Combining Results:**
Substituting back into the AM-GM inequality gives:
\[ \frac{\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}}{4} \geq 1 \]
Multiplying both sides by 4 results in:
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4 \]

5. **Conclusion:**
Since \(a\), \(b\), \(c\), and \(d\) are of the same sign, the expression \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}\) can take values greater than or equal to 4. Thus, the value of the expression lies in the interval:
\[ [4, \infty) \]

### Final Answer:
The value of \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}\) lies in the interval \([4, \infty)\).