91Ó°ÊÓ

In optimization problems, such as finding the maximum volume in a geometric problem, identifying critical points is essential. Critical points occur where the derivative of a function is zero or does not exist. These points may correspond to a maximum, minimum, or saddle point. To determine the critical points in our box problem, we first calculate the derivative of the volume function calculated: \(V = \frac{x}{2} - 4x^2 + 4x^3\).

The derivative \(V'(x) = \frac{1}{2} - 8x + 12x^2\) helps us assess where the rate of change (volume increase) of the function is zero. By setting \(V'(x) = 0\), we solve for \(x\) using the quadratic formula, which efficiently locates our critical points. Distinguishing between these points to find a maximum involves evaluating the volume function at each critical point, verifying the highest value within any given problem constraints.

The volume of a box is a classical concept in geometry and is calculated by multiplying the dimensions of the box — specifically its length, width, and height. In this case of an open-top box formed by folding a piece of cardboard, understanding this concept is critical for solving the exercise.

After cutting squares from each corner of the cardboard, the remaining portion is folded into a box, where:

• The length becomes \(1 - 2x\) meters
• The width becomes \(\frac{1}{2} - 2x\) meters
• The height is \(x\) meters

The formula for the volume \(V\) is given by \(V = \text{length} \times \text{width} \times \text{height} = (1-2x)(\frac{1}{2}-2x)x\).

This formula encapsulates the essence of the box's volume. Expanding and simplifying these terms allows us to find a manageable equation for optimization. Understanding how to alter dimensions by manipulating \(x\) can lead to finding the optimal size with the greatest volume.

The quadratic formula is a tool for solving equations of the form \(ax^2 + bx + c = 0\). It's particularly useful when simple factorization isn't feasible, such as in this exercise on optimizing the box's dimensions.

In the box example, once we derived the equation from the derivative \(12x^2 - 8x + \frac{1}{2} = 0\), we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula helps us determine the precise values for \(x\), which are the critical points of the volume function. Plugging in \( a = 12\), \( b = -8\), and \( c = \frac{1}{2}\) allows us to calculate the possible values of \(x\).

Just as importantly, understanding the solutions provided by the quadratic formula helps us deduce feasible values: checking values ensures they meet any additional constraints noted by the problem.