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The problem of finding the largest volume for a box that Vanilla Box Company can create from a 12x12-inch cardboard is a classic calculus optimization problem. When creating a box without a top from a rectangular piece of cardboard, it's important to consider what happens to the dimensions:

1. **Dimensions after Folding**: After cutting equal-sized squares from each corner, the cardboard is folded to form the sides of the box. If the side of each square cut out is defined as \(x\), then: - The new length and width each become \(12 - 2x\). - The height of the box is \(x\).
2. **Volume Formula**: Once folded into a box, the volume \(V\) of the box is a product of its length, width, and height: - \[ V(x) = (12-2x)(12-2x)x \] This function represents how the volume depends on the size \(x\) of the cut-out squares.

In conclusion, finding the volume involves understanding how removing and folding up the sides affect the original dimensions.

In solving the box problem, we use a quadratic equation within a cubic expression to determine the critical points. After expanding our volume function using multiplication, we derived:

\[ V(x) = 4x^3 - 48x^2 + 144x \]
These coefficients (4 for \(x^3\), -48 for \(x^2\), and 144 for \(x\)) are essential when finding where the function's derivative equals zero. In calculus, critical points happen where the first derivative equals zero or is undefined.

Using the derivative \(V'(x) = 12x^2 - 96x + 144\), we set it to zero and solve the resulting quadratic equation using the quadratic formula:

\[ x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} \]
For this equation, \(a = 12\), \(b = -96\), and \(c = 144\). By substituting these values, we find potential solutions for \(x\) which determine the sizes of cutout squares for maximum volume.

The derivative \(V'(x)\) of the volume function is central to locating the optimal size of the cut-out squares. By calculating \(V'(x)\), we find the rate of change of the volume with respect to \(x\). The derivative \(12x^2 - 96x + 144\) helps pinpoint where the function increases or decreases.

Steps to find critical points include:
- **Find Derivative**: Calculate the first derivative of \(V(x)\) to understand how volume changes. This step gives us \(V'(x) = 12x^2 - 96x + 144\).- **Set Derivative to Zero**: Solving the equation \(12x^2 - 96x + 144 = 0\) for \(x\) will yield potential critical points where the volume might be a maximum or minimum.

After solving, we get possible \(x\) values. These potential values are tested within the context of box dimensions (validity) to ensure they result in a box physically possible to create. This method effectively narrows down to the maximum volume configuration, ensuring \(V(x)\) is at its highest feasible value for the open-topped box.