Based on material learned earlier in the course. The purpose...

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine where the graph of $f(x)=x^{4}-5 x^{2}-6$ is below the graph of $g(x)=2 x^{2}+12$ by solving the inequality $f(x) \leq g(x)$.

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Determine where the graph of $f(x)=x^{4}-5 x^{2}-6$ is below the graph of $g(x)=2 x^{2}+12$ by solving the inequality $f(x) \leq g(x)$.

Key Concepts

Sign Analysis and Interval Testing

After factoring a polynomial, sign analysis involves determining the sign (positive or negative) of the expression in each interval defined by its critical points. This step is crucial for identifying the intervals that satisfy the inequality. By choosing test points from each interval, one can conclude where the expression meets the given inequality condition.

Factoring Polynomials

Factoring is a key tool used to solve polynomial equations and inequalities. By expressing a polynomial as a product of simpler factors, one can identify its roots or critical points. These factors correspond to points where the function equals zero, which are essential for determining the intervals to test in a sign chart analysis.

Graphical Interpretation of Functions

This concept focuses on understanding how the graphs of functions relate to each other. When analyzing where one function is below another, the inequality corresponds to points where the function values of the first are less than or equal to those of the second. This interpretation connects algebraic solutions of inequalities with their geometric representations on a coordinate graph.

Polynomial Inequalities

This concept involves solving inequalities where one side is a polynomial expression. It requires finding the values for which the polynomial is either greater than, less than, or equal to zero. The process generally includes rewriting the inequality in a form where one side is zero, allowing for the determination of critical points (or roots) that partition the number line into intervals to be analyzed.

Transforming an Inequality by Subtraction

When comparing two functions to determine where one is below the other, it is common to subtract one function from the other to form a single inequality. This technique reduces the problem to finding where the resulting expression is less than or equal to zero. It simplifies analysis, as it converts a relational statement between two functions into a standard polynomial inequality.

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