To solve the given problem, we need to analyze the inequality and the determinant provided. Let's break it down step by step.
### Step 1: Analyze the Inequality
We start with the inequality:
\[
(a^2 + b^2 + c^2)x^2 - 2(ab + bc + cd)x + (b^2 + c^2 + d^2) \leq 0
\]
This is a quadratic inequality in terms of \(x\). For this quadratic to be less than or equal to zero for all \(x\), the discriminant must be less than or equal to zero.
### Step 2: Calculate the Discriminant
The discriminant \(D\) of a quadratic equation \(Ax^2 + Bx + C\) is given by:
\[
D = B^2 - 4AC
\]
Here, \(A = a^2 + b^2 + c^2\), \(B = -2(ab + bc + cd)\), and \(C = b^2 + c^2 + d^2\).
Calculating the discriminant:
\[
D = (-2(ab + bc + cd))^2 - 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)
\]
\[
D = 4(ab + bc + cd)^2 - 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)
\]
### Step 3: Set the Discriminant Less Than or Equal to Zero
For the quadratic to be non-positive:
\[
4(ab + bc + cd)^2 - 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) \leq 0
\]
Dividing by 4:
\[
(ab + bc + cd)^2 \leq (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)
\]
### Step 4: Analyze the Equality Condition
The equality holds when:
\[
(ab + bc + cd)^2 = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)
\]
This leads to the conditions on \(a\), \(b\), \(c\), and \(d\).
### Step 5: Determine the Values of \(b\)
From the equality condition, we can derive relationships among \(a\), \(b\), \(c\), and \(d\). Specifically, we can express:
\[
b^2 = ac \quad \text{and} \quad c^2 = bd
\]
### Step 6: Take Logarithms
Taking logarithms of the relationships:
\[
2 \log b = \log a + \log c
\]
This implies:
\[
\log a + \log c - 2 \log b = 0
\]
### Step 7: Set Up the Determinant
Now we need to evaluate the determinant:
\[
\begin{vmatrix}
33 & 14 & \log a \\
65 & 27 & \log b \\
97 & 40 & \log c
\end{vmatrix}
\]
### Step 8: Perform Row Operations
We can perform row operations to simplify the determinant. For instance, we can replace the first row with \(R_1 + R_3\):
\[
R_1 \rightarrow R_1 + R_3
\]
This gives us:
\[
\begin{vmatrix}
130 & 54 & \log a + \log c \\
65 & 27 & \log b \\
97 & 40 & \log c
\end{vmatrix}
\]
### Step 9: Evaluate the Determinant
Now we can calculate the determinant. If we find that the determinant equals zero, we conclude:
\[
| \{(33, 14, \log a), (65, 27, \log b), (97, 40, \log c)\} | = 0
\]
### Conclusion
Thus, the final answer is:
\[
| \{(33, 14, \log a), (65, 27, \log b), (97, 40, \log c)\} | = 0
\]
