lf `abcd=1` where a,b,c,d are positive reals then the minimum value of `a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd` is

Explanation

To find the minimum value of the expression \( S = a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + cd \) given that \( abcd = 1 \) where \( a, b, c, d \) are positive reals, we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We need to minimize the expression \( S \). It consists of the squares of the variables and the products of pairs of these variables.

2. **Applying AM-GM Inequality**:
We can apply the AM-GM inequality to the terms of \( S \). The AM-GM inequality states that for any non-negative 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.

3. **Grouping Terms**:
We can group the terms in \( S \) as follows:
\[ S = (a^2 + b^2 + c^2 + d^2) + (ab + ac + ad + bc + bd + cd) \]
We will apply AM-GM to both groups.

4. **Applying AM-GM to the Squares**:
For the squares:
\[ \frac{a^2 + b^2 + c^2 + d^2}{4} \geq \sqrt[4]{a^2b^2c^2d^2} = \sqrt[4]{(abcd)^2} = \sqrt[4]{1^2} = 1 \]
Thus,
\[ a^2 + b^2 + c^2 + d^2 \geq 4 \]

5. **Applying AM-GM to the Products**:
For the products:
\[ \frac{ab + ac + ad + bc + bd + cd}{6} \geq \sqrt[6]{(ab)(ac)(ad)(bc)(bd)(cd)} \]
We can simplify the right-hand side:
\[ (ab)(ac)(ad)(bc)(bd)(cd) = a^3b^3c^3d^3 = (abcd)^{3} = 1^3 = 1 \]
Thus,
\[ ab + ac + ad + bc + bd + cd \geq 6 \]

6. **Combining Results**:
Now we combine the results from the two applications of AM-GM:
\[ S \geq (a^2 + b^2 + c^2 + d^2) + (ab + ac + ad + bc + bd + cd) \geq 4 + 6 = 10 \]

7. **Finding Equality Condition**:
The equality in AM-GM holds when all the terms are equal. This means:
\[ a = b = c = d \]
Given \( abcd = 1 \), if \( a = b = c = d = k \), then \( k^4 = 1 \) implies \( k = 1 \). Therefore, \( a = b = c = d = 1 \).

8. **Conclusion**:
The minimum value of \( S \) is achieved when \( a = b = c = d = 1 \):
\[ S = 1^2 + 1^2 + 1^2 + 1^2 + 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 4 + 6 = 10 \]

Thus, the minimum value of \( S \) is \( \boxed{10} \).