Question

which table describes the behavior of the graph of \\(f(x) = 2x^3 - 26x - 24\\)?

\

$$\begin{tabular}{|c|c|} \\hline \\textbf{interval} & \\textbf{relation of graph to x-axis} \\\\ \\hline \\((-\\infty, -3)\\) & above \\\\ \\hline \\((-3, -1)\\) & below \\\\ \\hline \\((-1, 4)\\) & above \\\\ \\hline \\((4, \\infty)\\) & below \\\\ \\hline \\end{tabular}$$

\

$$\begin{tabular}{|c|c|} \\hline \\textbf{interval} & \\textbf{relation of graph to x-axis} \\\\ \\hline \\((-\\infty, -3)\\) & below \\\\ \\hline \\((-3, -1)\\) & above \\\\ \\hline \\((-1, 4)\\) & below \\\\ \\hline \\((4, \\infty)\\) & above \\\\ \\hline \\end{tabular}$$

\

$$\begin{tabular}{|c|c|} \\hline \\textbf{interval} & \\textbf{relation of graph to x-axis} \\\\ \\hline \\((-\\infty, -4)\\) & above \\\\ \\hline \\((-4, 1)\\) & below \\\\ \\hline \\((1, 3)\\) & above \\\\ \\hline \\((3, \\infty)\\) & below \\\\ \\hline \\end{tabular}$$

Explanation:

Find the roots of the polynomial

Using the Rational Root Theorem and Polynomial Factoring knowledge points

$$
LATEXBLOCK0
$$

The roots are \(x = -3\), \(x = -1\), and \(x = 4\).

Determine the sign in each interval

We analyze the sign of \(f(x)\) on the intervals defined by the roots:

• For \((-\infty, -3)\): Choose \(x = -4\).

$$f(-4) = 2(-1)(-3)(-8) < 0 \quad \text{(Below)}$$

• For \((-3, -1)\): Choose \(x = -2\).

$$f(-2) = 2(1)(-1)(-6) > 0 \quad \text{(Above)}$$

• For \((-1, 4)\): Choose \(x = 0\).

$$f(0) = -24 < 0 \quad \text{(Below)}$$

• For \((4, \infty)\): Choose \(x = 5\).

$$f(5) = 2(8)(6)(1) > 0 \quad \text{(Above)}$$

Match with the correct table option

Comparing our sign analysis to the given choices:

• \((-\infty, -3)\): Below
• \((-3, -1)\): Above
• \((-1, 4)\): Below
• \((4, \infty)\): Above

This matches the second table option.

Answer:

• (A) First table: Above, Below, Above, Below for intervals \((-\infty, -3)\), \((-3, -1)\), \((-1, 4)\), \((4, \infty)\)
• (B) Second table: Below, Above, Below, Above for intervals \((-\infty, -3)\), \((-3, -1)\), \((-1, 4)\), \((4, \infty)\) (Correct answer)
• (C) Third table: Above, Below, Above, Below for intervals \((-\infty, -4)\), \((-4, 1)\), \((1, 3)\), \((3, \infty)\)