To find the intersection points of the given function $y = \frac{x^3 - 1}{4x^2}$ with the $OX$-axis, we need to determine where $y = 0$. Let us solve this step-by-step:

Step 1: Set $y = 0$

$\frac{x^3 - 1}{4x^2} = 0$

The numerator must be zero for the fraction to be zero: $x^3 - 1 = 0$

Step 2: Solve for $x$

$x^3 = 1 \quad \Rightarrow \quad x = 1$

Step 3: Check the denominator

The denominator, $4x^2$, should not be zero. Therefore, $x \neq 0$.

Step 4: Coordinates of the point

When $x = 1$, $y = 0$. Hence, the intersection point is: $1,0$

Final Answer:

$1,0$

Would you like further details or clarification?

Questions to Expand Understanding:

1. How do we analyze intersections with the $OY$-axis for this function?
2. What happens to the function as $x \to 0^+$ and $x \to 0^-$?
3. What is the domain of this function?
4. How would the graph look in a neighborhood around $x = 0$?
5. What are the asymptotes (if any) for this function?

Tip:

Always check the domain of a function before solving for zeros, as undefined points like $x = 0$ can impact your results.