I want to calculate the minimum point of the function f(x)= (a^(1/2) ).(3.a.(x^2)-(a^2 ).x+(a^3) )/((2.√((x^2)+(d^2) )).(√k)) as so as the corresponding minimum value of the function

To find the minimum point of the function \( f(x) \) and the corresponding minimum value, we need to follow these steps:

1. **Define the function**: 
   \[
   f(x) = \frac{\sqrt{a} \cdot (3a x^2 - a^2 x + a^3)}{2 \sqrt{x^2 + d^2} \cdot \sqrt{k}}
   \]

2. **Find the derivative** of \( f(x) \) with respect to \( x \) (denoted as \( f'(x) \)).

3. **Set the derivative equal to zero** to find the critical points:
   \[
   f'(x) = 0
   \]
   This equation will help us find the \( x \)-value(s) where the function might have a minimum or maximum.

4. **Determine the second derivative** \( f''(x) \) to confirm whether the critical points are minimum or maximum:
   \[
   f''(x) > 0 \quad \text{(Minimum point)}
   \]
   \[
   f''(x) < 0 \quad \text{(Maximum point)}
   \]

5. **Substitute the critical points into \( f(x) \)** to find the corresponding minimum value.

Let's proceed with the differentiation and critical point analysis.

### Step 1: The Function
\[
f(x) = \frac{\sqrt{a} \cdot (3a x^2 - a^2 x + a^3)}{2 \sqrt{x^2 + d^2} \cdot \sqrt{k}}
\]

### Step 2: First Derivative \( f'(x) \)
We apply the quotient rule for differentiation, which is given by:
\[
f'(x) = \frac{(v \cdot u' - u \cdot v')}{v^2}
\]
where \( u(x) = \sqrt{a} \cdot (3a x^2 - a^2 x + a^3) \) and \( v(x) = 2 \sqrt{x^2 + d^2} \cdot \sqrt{k} \).

I'll calculate the derivative and solve for \( f'(x) = 0 \).

### Step 3: Find Critical Points
Setting \( f'(x) = 0 \) gives us the equation to solve for \( x \).

### Step 4: Second Derivative \( f''(x) \)
We'll take the derivative of \( f'(x) \) to confirm whether the critical point is a minimum.

### Step 5: Minimum Value
Once we have \( x \), we substitute it back into the original function \( f(x) \) to find the corresponding minimum value.

Let's start by differentiating \( f(x) \) with respect to \( x \) to find \( f'(x) \). This requires some computation, so I will calculate it step by step.The first derivative of the function \( f(x) \) with respect to \( x \) is:

\[
f'(x) = \frac{a^{3/2} \cdot (-a^2 x - a d^2 + 6 d^2 x + 3x^3)}{2 \sqrt{k} \cdot (d^2 + x^2)^{3/2}}
\]

### Step 3: Find the Critical Points

To find the critical points, we set \( f'(x) = 0 \) and solve for \( x \):

\[
\frac{a^{3/2} \cdot (-a^2 x - a d^2 + 6 d^2 x + 3x^3)}{2 \sqrt{k} \cdot (d^2 + x^2)^{3/2}} = 0
\]

The numerator must equal zero for the entire expression to be zero:

\[
-a^2 x - a d^2 + 6 d^2 x + 3x^3 = 0
\]

I'll solve this cubic equation for \( x \) to find the critical points.The critical points of the function are complex expressions:

\[
x_1 = \frac{2^{1/3} \left(2a^2 - 12d^2 + 2^{1/3} \left(9ad^2 + \sqrt{81a^2d^4 - 4(a^2 - 6d^2)^3}\right)^{2/3}\right)}{6\left(9ad^2 + \sqrt{81a^2d^4 - 4(a^2 - 6d^2)^3}\right)^{1/3}}
\]

The other two critical points involve complex numbers, which might not be physically meaningful depending on the context.

### Step 4: Determine if it's a Minimum

Next, we can compute the second derivative \( f''(x) \) and substitute \( x_1 \) to confirm whether it's a minimum point. 

If \( f''(x_1) > 0 \), then \( x_1 \) is a minimum point.

Would you like me to compute \( f''(x) \) and evaluate it at \( x_1 \), or would you prefer more details on a specific part?

Learn how to calculate the minimum point and corresponding value of the function f(x) = (sqrt(a) * (3ax^2 - a^2x + a^3)) / (2 * sqrt(x^2 + d^2) * sqrt(k)). This involves differentiating the function, finding critical points, and evaluating the second derivative to confirm the minimum. Suitable for advanced mathematics enthusiasts.

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