If `ax^(3)-cx+bge0AAxepsilonR^(+)-{0}` where `a,b,cepsilonR^(+)`. Then the minimum value of `((27ab^(2))/(c^(3)))` is………

Explanation

To solve the problem, we need to find the minimum value of the expression \(\frac{27ab^2}{c^3}\) given the condition \(ax^3 - cx + b \geq 0\) for \(x \in \mathbb{R}^+\) and \(a, b, c \in \mathbb{R}^+\).

### Step-by-step Solution:

1. **Understanding the Condition**:
The condition \(ax^3 - cx + b \geq 0\) implies that the cubic function \(f(x) = ax^3 - cx + b\) must be non-negative for all positive \(x\).

2. **Finding the Critical Points**:
To find the critical points, we differentiate \(f(x)\):
\[ f'(x) = 3ax^2 - c \]
Setting \(f'(x) = 0\) gives:
\[ 3ax^2 - c = 0 \implies x^2 = \frac{c}{3a} \implies x = \sqrt{\frac{c}{3a}} \]
(We only consider the positive root since \(x \in \mathbb{R}^+\)).

3. **Evaluating the Function at the Critical Point**:
Now, we evaluate \(f\) at \(x = \sqrt{\frac{c}{3a}}\):
\[ f\left(\sqrt{\frac{c}{3a}}\right) = a\left(\sqrt{\frac{c}{3a}}\right)^3 - c\left(\sqrt{\frac{c}{3a}}\right) + b \]
Simplifying this:
\[ = a\left(\frac{c^{3/2}}{(3a)^{3/2}}\right) - c\left(\sqrt{\frac{c}{3a}}\right) + b \]
\[ = \frac{ac^{3/2}}{3\sqrt{3}a^{3/2}} - \frac{c^{3/2}}{\sqrt{3a}} + b \]
\[ = \frac{c^{3/2}}{3\sqrt{3}a^{1/2}} - \frac{c^{3/2}}{\sqrt{3a}} + b \]
\[ = \frac{c^{3/2}}{3\sqrt{3}a^{1/2}} - \frac{3c^{3/2}}{3\sqrt{3a}} + b \]
\[ = \frac{c^{3/2}}{3\sqrt{3}a^{1/2}} - \frac{3c^{3/2}}{3\sqrt{3a}} + b \]
\[ = b - \frac{2c^{3/2}}{3\sqrt{3a}} \]

4. **Setting the Condition for Non-negativity**:
For \(f\left(\sqrt{\frac{c}{3a}}\right) \geq 0\):
\[ b - \frac{2c^{3/2}}{3\sqrt{3a}} \geq 0 \implies b \geq \frac{2c^{3/2}}{3\sqrt{3a}} \]

5. **Substituting into the Expression**:
We want to minimize \(\frac{27ab^2}{c^3}\). Using the inequality from step 4:
\[ b^2 \geq \left(\frac{2c^{3/2}}{3\sqrt{3a}}\right)^2 = \frac{4c^3}{27a} \]
Thus,
\[ \frac{27ab^2}{c^3} \geq \frac{27a \cdot \frac{4c^3}{27a}}{c^3} = 4 \]

6. **Conclusion**:
Therefore, the minimum value of \(\frac{27ab^2}{c^3}\) is \(\boxed{4}\).