If the cubic polymomial `y = ax ^(3) + bx^(2) +cx+ d (a,b,c,d inR)` has only one critical point in its entire domain and `ac=2,` then the value of `|b|` is:

Options

A. `sqrt2`
B. `sqrt3`
C. `sqrt5`
D. `sqrt6`

Answer

To solve the problem, we need to find the value of \(|b|\) for the cubic polynomial given the conditions stated. Here’s the step-by-step solution:

### Step 1: Write the cubic polynomial
The cubic polynomial is given as:
\[ y = ax^3 + bx^2 + cx + d \]
where \(a, b, c, d \in \mathbb{R}\).

### Step 2: Differentiate the polynomial
To find the critical points, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = 3ax^2 + 2bx + c \]

### Step 3: Set the derivative to zero
For critical points, we set the derivative equal to zero:
\[ 3ax^2 + 2bx + c = 0 \]
This is a quadratic equation in \(x\).

### Step 4: Condition for one critical point
The cubic polynomial has only one critical point if the quadratic equation has only one root. This occurs when the discriminant \(D\) of the quadratic equation is zero:
\[ D = (2b)^2 - 4(3a)(c) = 0 \]

### Step 5: Calculate the discriminant
Calculating the discriminant:
\[ D = 4b^2 - 12ac = 0 \]
Rearranging gives:
\[ 4b^2 = 12ac \]
\[ b^2 = 3ac \]

### Step 6: Substitute the given condition
We are given that \(ac = 2\). Substituting this into the equation:
\[ b^2 = 3 \cdot 2 = 6 \]

### Step 7: Solve for \(b\)
Taking the square root of both sides gives:
\[ b = \pm \sqrt{6} \]

### Step 8: Find \(|b|\)
The absolute value of \(b\) is:
\[ |b| = \sqrt{6} \]

### Final Answer
Thus, the value of \(|b|\) is:
\[ \boxed{\sqrt{6}} \]