Problem 55 - Calculus with Concepts in Calculus

Question

Show that the given inequality holds for the given values of \(\mathrm{x}\). $$ x^{4}-4 x>-3 \text { for } x>1 $$

Step-by-Step Solution

Answer

For \(x > 1\), the inequality holds because \(x^4\) grows quickly, making \(x^4 - 4x + 3 > 0\).

Step 1: Analyze the Inequality

The given inequality is \(x^4 - 4x > -3\). This can also be rewritten as \(x^4 - 4x + 3 > 0\).

Step 2: Rewrite the Inequality as a Polynomial Equation

Rewrite the inequality as: \[ x^4 - 4x + 3 = 0 \] This step is to find the critical points where the inequality may change sign.

Step 3: Solve for Roots

Find the roots of the polynomial \( x^4 - 4x + 3 = 0 \) to determine where the expression changes sign. This may involve trying possible rational roots or factoring if possible.

Step 4: Determine the Behavior of the Polynomial

Without roots solved, use the factored form or derivative analysis to determine where \(x^4 - 4x + 3\) is positive. Over \(x > 1\), substitute values or analyze sign changes in segments.

Step 5: Test the Inequality Over the Interval \(x > 1\)

Test values from \(x > 1\) into \(x^4 - 4x + 3\). Generally, polynomial growth suggests positivity as \(x\) increases, indicating satisfaction of the inequality.

Step 6: Conclude the Inequality's Validity

For \(x > 1\), the term \(x^4\) grows quickly outstripping \(-4x + 3\), ensuring \(x^4 - 4x + 3 > 0\). Thus, the inequality holds true for \(x > 1\).

Key Concepts

Inequality Analysis

When addressing polynomial inequalities, like the problem given, we start by analyzing the inequality's structure. Let's consider the transformation from the original expression: - Original Expression: \(x^4 - 4x > -3\) - Transformed Expression: \(x^4 - 4x + 3 > 0\) The goal is to determine where this expression holds true, specifically for \(x > 1\). Key steps involve:

• Identifying how to transform the inequality into a standard polynomial form, as seen in this example from \(x^4 - 4x > -3\) to \(x^4 - 4x + 3 > 0\).
• Understanding that solutions involve checking intervals where the inequality is satisfied, considering the polynomial's behavior across its domain.

These steps are crucial as they guide us to identify value ranges that satisfy the original inequality, allowing a deeper understanding of how a polynomial relates to greater or lesser conditions.

Critical Points in Polynomials

Critical points in polynomials, such as those in the problem \(x^4 - 4x + 3 > 0\), are vital for understanding the polynomial's behavior. They include points where the polynomial equals zero or where its derivative is zero. These points help determine where the graph changes its pattern. To find the critical points, you need to set the polynomial equal to zero, resulting in the equation: \[x^4 - 4x + 3 = 0\]Finding these roots, though challenging, helps understand where the polynomial might cross the x-axis or where its slope changes. The identification of critical points aids in:

• Identifying regions of the domain where the polynomial might switch signs.
• Providing insight into possible intervals to test the inequality's satisfaction.
• Helping visualize where the polynomial might be increasing or decreasing.

Recognizing these points is foundational for solving polynomial inequalities effectively, as it informs both the algebraic and graphical analysis of the inequality.

Polynomial Roots

The process of identifying polynomial roots is a key aspect of solving polynomial inequalities like \(x^4 - 4x + 3 = 0\). Roots of a polynomial are the solutions where the polynomial evaluates to zero. These roots demarcate intervals where the polynomial changes sign, essential for determining inequality satisfaction. Finding roots involves several strategies:

• Checking for simple factorization if possible.
• Utilizing methods such as the Rational Root Theorem to test for potential rational solutions.
• Employing numerical or iterative methods if algebraic strategies are not easily applicable.

In our example, knowing the roots allows us to examine segments on the number line—where the sign of \(x^4 - 4x + 3\) shifts. This understanding directly informs us whether the polynomial satisfies the inequality in given intervals, like \(x > 1\). Through these methods, students appreciate how roots are not merely intersection points but are integral to inequality analysis.