The task involves analyzing the function $y = \frac{3|x|}{2x^2 + x - 3}$. Here's what needs to be done:

1. Investigate the continuity of the function:

   • Identify points of discontinuity by analyzing the denominator $2x^2 + x - 3$, as division by zero is undefined.
   • Study the behavior of $|x|$ and its influence on the function's continuity.

2. Determine the asymptotes:

   • Vertical asymptotes occur at the roots of $2x^2 + x - 3 = 0$, if the numerator does not also vanish at these points.
   • Horizontal or oblique asymptotes can be analyzed by evaluating the behavior of $y$ as $x \to \infty$ or $x \to -\infty$.

3. Plot a rough sketch of the function:

   • Consider key points (intercepts and asymptotes).
   • Analyze the sign of $y$ in different intervals based on $2x^2 + x - 3$ and $|x|$.

4. Find and classify discontinuities:

   • For points where $y$ is undefined, determine whether the discontinuity is removable or non-removable.

5. Formulate conditions for extrema:

   • Compute $y'$, the derivative of $y$, to find critical points.
   • Determine the necessary and sufficient conditions for extrema based on $y'(x) = 0$ or $y'(x)$ being undefined.

Would you like me to proceed with solving these steps in detail?