91Ó°ÊÓ

Critical points are important numbers where the values of functions could potentially change direction. For inequality problems like this one, critical points occur where the expression changes from less than zero to greater than zero, or vice versa. To find the critical points for the equation \(x^{4}(x-3) \leq 0\), set the inequality equal to zero. This gives us equations \(x^4 = 0\) and \(x - 3 = 0\). Solving these, we discover the critical points are \(x = 0\) and \(x = 3\). These are the points where the inequality may change behavior because they make the expression equal to zero.

Once you have these critical points, they can help to divide the number line into manageable sections for further analysis.

Interval notation is a system of expressing a range of numbers where solutions to an inequality exist. It's a neat way to describe all possible solutions in condensed form. In the example, the critical points \(x = 0\) and \(x = 3\) partition the number line into three intervals:

• \((- \infty, 0)\)
• \((0, 3)\)
• \((3, \infty)\)

Each interval represents a potential set of solutions based on whether plugging values from that interval into the inequality returns a true statement. We keep endpoints in the intervals using square brackets like "\([0, 3]\)" when those points make the inequality true. Here, the combined solution is \((- \infty, 3]\).

Always pay attention to whether to include endpoints in the intervals based on the direction of the inequality.

Once you have critical points and have broken the number line into intervals, we use "testing intervals" to determine where the inequality holds true. Choose any test point from each interval to substitute back into the inequality. In this solution:

• Choose \(-1\) from \((- \infty, 0)\)
• Choose \(1\) from \((0, 3)\)
• Choose \(4\) from \((3, \infty)\)

Substituting each of these into the inequality \(x^{4}(x - 3) \leq 0\), we find:

• For \(-1\), the result is negative, which satisfies the inequality.
• For \(1\), the product is still negative, confirming this as a solution interval.
• For \(4\), the product is positive, so this interval doesn't satisfy the inequality.

This process ensures that intervals that make the inequality true are correctly identified, which we use to express the solution as \((- \infty, 3]\).