Let `A=[(2,b,1),(b,b^(2)+1,b),(1,b,2)]` where `b gt 0`. Then the minimum value of `("det.(A)")/(b)` is

Options

A. `sqrt(3)`
B. `-sqrt(3)`
C. `-2sqrt(3)`
D. `2sqrt(3)`

Answer

To solve the problem, we need to find the minimum value of \(\frac{\text{det}(A)}{b}\) where \(A = \begin{pmatrix} 2 & b & 1 \\ b & b^2 + 1 & b \\ 1 & b & 2 \end{pmatrix}\) and \(b > 0\).

### Step 1: Calculate the Determinant of Matrix A

We will calculate the determinant of matrix \(A\) using the formula for the determinant of a \(3 \times 3\) matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \(A\):

\[ \text{det}(A) = 2 \left( (b^2 + 1) \cdot 2 - b \cdot b \right) - b \left( b \cdot 2 - b \cdot 1 \right) + 1 \left( b \cdot b - (b^2 + 1) \cdot 1 \right) \]

Calculating each term:

1. First term:
\[ 2 \left( (b^2 + 1) \cdot 2 - b^2 \right) = 2 \left( 2b^2 + 2 - b^2 \right) = 2( b^2 + 2) = 2b^2 + 4 \]

2. Second term:
\[ -b \left( 2b - b \right) = -b^2 \]

3. Third term:
\[ 1 \left( b^2 - (b^2 + 1) \right) = 1 \left( b^2 - b^2 - 1 \right) = -1 \]

Combining these results:
\[ \text{det}(A) = (2b^2 + 4) - b^2 - 1 = b^2 + 3 \]

### Step 2: Formulate the Expression to Minimize

We need to minimize:
\[ \frac{\text{det}(A)}{b} = \frac{b^2 + 3}{b} = b + \frac{3}{b} \]

### Step 3: Find the Critical Points

To find the minimum value, we differentiate \(f(b) = b + \frac{3}{b}\):

\[ f'(b) = 1 - \frac{3}{b^2} \]

Setting the derivative to zero to find critical points:
\[ 1 - \frac{3}{b^2} = 0 \implies \frac{3}{b^2} = 1 \implies b^2 = 3 \implies b = \sqrt{3} \]

### Step 4: Verify Minimum Value

To confirm it's a minimum, we check the second derivative:
\[ f''(b) = \frac{6}{b^3} \]
Since \(b > 0\), \(f''(b) > 0\) indicates that \(f(b)\) is concave up at \(b = \sqrt{3}\), confirming a local minimum.

### Step 5: Calculate the Minimum Value

Substituting \(b = \sqrt{3}\) into \(f(b)\):
\[ f(\sqrt{3}) = \sqrt{3} + \frac{3}{\sqrt{3}} = \sqrt{3} + \sqrt{3} = 2\sqrt{3} \]

Thus, the minimum value of \(\frac{\text{det}(A)}{b}\) is:
\[ \boxed{2\sqrt{3}} \]