Let $f(x)=A x-\frac{B}{x^{3}}$, where $A$ and $B$ are real numbers. Which of the following is a critical value of $f(x)$ ?

Hint: You may assume that $A$ and $B$ are values such that the correct answer is defined.
$\sqrt[4]{\frac{B}{3 A}}$
$\sqrt[4]{-\frac{B}{A}}$
$\sqrt[4]{-\frac{3 B}{A}}$
$\sqrt[4]{\frac{3 B}{A}}$
$\sqrt[4]{\frac{B}{A}}$
$\sqrt[4]{-\frac{B}{3 A}}$

Step 1 ：Given the function \(f(x)=Ax-\frac{B}{x^{3}}\), we need to find the critical points. The critical points of a function are the points where the derivative of the function is either zero or undefined.

Step 2 ：First, we find the derivative of the function \(f(x)\). The derivative of \(f(x)\) is \(f'(x)=A+\frac{3B}{x^{4}}\).

Step 3 ：Next, we set the derivative equal to zero and solve for \(x\). This gives us the critical points of the function.

Step 4 ：The solutions to the equation \(f'(x)=0\) are \(-3^{1/4}(-B/A)^{1/4}\), \(3^{1/4}(-B/A)^{1/4}\), \(-3^{1/4}i(-B/A)^{1/4}\), and \(3^{1/4}i(-B/A)^{1/4}\).

Step 5 ：However, the question asks for real numbers, so we can ignore the complex solutions. Therefore, the critical points are \(-3^{1/4}(-B/A)^{1/4}\) and \(3^{1/4}(-B/A)^{1/4}\).

Step 6 ：Final Answer: \(\boxed{-\sqrt[4]{\frac{3 B}{A}}}\) and \(\boxed{\sqrt[4]{\frac{3 B}{A}}}\)