Problem 32 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ 8 x+\frac{1}{x^{2}}<0 $$

Step-by-Step Solution

Answer

The solution is \(x \in (-\infty, -\frac{1}{2})\).

Step 1: Understand the inequality

We need to solve the inequality \(8x + \frac{1}{x^2} < 0\). This means we are looking for the values of \(x\) that make this expression negative. Start by identifying the problematic points where the expression might not be defined, particularly when \(x = 0\) because of the term \(\frac{1}{x^2}\).

Step 2: Find critical points

Set the inequality to zero to find the critical points: \(8x + \frac{1}{x^2} = 0\). This can be rewritten as \(8x = -\frac{1}{x^2}\). Multiplying both sides by \(x^2\) gives \(8x^3 = -1\). Solving for \(x\), we find the critical point: \(x = -\sqrt[3]{\frac{1}{8}} = -\frac{1}{2}\).

Step 3: Consider intervals

Critical points and undefined points (\(x = 0\)) split the number line into intervals: \((-\infty, -\frac{1}{2})\), \((-\frac{1}{2}, 0)\), and \((0, \infty)\). We need to test the sign of \(8x + \frac{1}{x^2}\) in each interval.

Step 4: Test intervals

Choose a test point from each interval: - For \((-\infty, -\frac{1}{2})\), test \(x = -1\): \(8(-1) + \frac{1}{(-1)^2} = -8 + 1 = -7 < 0\).- For \((-\frac{1}{2}, 0)\), test \(x = -\frac{1}{4}\): \(8(-\frac{1}{4}) + \frac{1}{(-\frac{1}{4})^2} = -2 + 16 = 14 > 0\). Hence, this interval does not satisfy the inequality.- For \((0, \infty)\), test \(x = 1\): \(8(1) + \frac{1}{1^2} = 8 + 1 = 9 > 0\). Hence, this interval does not satisfy the inequality.

Step 5: Conclusion

From testing the intervals, we find that the inequality \(8x + \frac{1}{x^2} < 0\) holds in the interval \((-\infty, -\frac{1}{2})\). We exclude \(-\frac{1}{2}\) from the solution as it makes the expression zero, not negative.

Key Concepts

Critical Points

When dealing with inequalities in calculus, finding critical points is essential. Critical points occur where the derivative of a function equals zero or is undefined. They help us understand where a function may change its behavior, from increasing to decreasing or vice versa. In the context of solving inequalities such as \(8x + \frac{1}{x^2} < 0\), we seek points where the expression may be zero (turning point) or undefined (possible interruption).
By setting the inequality to zero, \(8x + \frac{1}{x^2} = 0\), we transform it into an equation. To clear the fraction, multiply through by \(x^2\), giving \(8x^3 = -1\). Solving this for \(x\), we find \(x = -\sqrt[3]{\frac{1}{8}} = -\frac{1}{2}\). This is our critical point, a key location where the inequality may change sign. Understanding critical points like this helps us identify intervals to test for solving the inequality.

Interval Testing

Interval testing is a powerful strategy to analyze inequalities and determine where they hold true. Once we have our critical points, which are \(-\frac{1}{2}\) and the undefined point, \(x = 0\), we can use these to divide the real number line into distinct intervals.

• First, interval \((-\infty, -\frac{1}{2})\).
• Second, interval \((-\frac{1}{2}, 0)\).
• Third, interval \((0, \infty)\).

To test an interval, select a number (test point) from within and substitute it back into the original inequality \(8x + \frac{1}{x^2}\). The sign of the result tells us whether the inequality holds in that entire interval:
- For the interval \((-\infty, -\frac{1}{2})\), use \(x = -1\) yielding \(-7\), which is less than 0, implying the inequality holds here.
- In \((-\frac{1}{2}, 0)\), using \(x = -\frac{1}{4}\) gives \(14\), which is greater than 0, meaning the inequality fails to hold.
- Finally, in \(0, \infty\), testing \(x = 1\) gives \(9\), again not satisfying \(8x + \frac{1}{x^2} < 0\).
Through testing, we found \((-\infty, -\frac{1}{2})\) as the interval where the inequality is true.

Undefined Expressions

Certain expressions can be undefined for specific values of \(x\). This is often due to division by zero, which in our exercise appears in the form \(\frac{1}{x^2}\). Here, when \(x = 0\), the expression becomes undefined since division by zero is not possible.
Identifying such undefined points is important in interval testing because they indicate where an expression may be discontinuous, leading to changes in behavior across intervals. In our example, this helps us break the number line into meaningful sections to be tested. An undefined point like \(x = 0\) serves as a boundary, impacting the evaluation of each relevant interval from our tested points.

• Carefully analyze around undefined points, as they mark transitions in the behavior of inequalities.
• Ensure that undefined regions are considered due to their mathematical significance in solving inequalities.

Overall, understanding how undefined expressions affect inequalities is crucial to a robust analysis, ensuring that solutions are comprehensive and accurately reflect the conditions of the problem.