Which table describes the behavior of the graph of f(x) = 2x³ - 26x - 24?

Solution:

Given, f(x) = 2x³ - 26x - 24

We have to find the table which describes the behaviour of the graph.

Now, 2x³ - 26x - 24 = 0

Splitting the middle term

2x³ - 2x - 24x - 24 = 0

Taking out the common terms

2x(x² - 1) - 24(x + 1) = 0

Using the algebraic identity a² - b² = (a + b) (a - b)

2x(x - 1)(x + 1) - 24(x + 1) = 0

(x + 1)(2x(x - 1) - 24) = 0

Using the distributive property

(x + 1)(2x² - 2x - 24) = 0

By taking 2 as common

(x + 1)2(x² - x - 12) = 0

(x + 1) (x² - x - 12) = 0

By splitting the middle term

(x + 1)(x² - 4x + 3x - 12) = 0

(x + 1) (x(x - 4) + 3(x - 4)) = 0

(x + 1) (x - 4) (x + 3) = 0

Now, x + 1 = 0

x = -1

x - 4 = 0

x = 4

x + 3 = 0

x = -3

Therefore, for \(x\in (-\infty , -3)\, y < 0\), below x-axis

\(x\in (-3, -1)\, y > 0,\), above x-axis

\(x\in (-1, 4)\, y < 0\), below x-axis

\(x\in (4,\infty )\, y > 0\), above x-axis.

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Which table describes the behavior of the graph of f(x) = 2x³ - 26x - 24?

Summary:

The behavior of the graph of f(x) = 2x³ - 26x - 24 is

For \(x\in (-\infty , -3)\, y < 0,\), below x-axis

\(x\in (-3, -1)\, y > 0,\), above x-axis

\(x\in (-1, 4)\, y < 0\), below x-axis

\(x\in (4,\infty )\, y > 0\), above x-axis.