use the intermediate value theorem to determine whether the function $f(x) = x^3 + 2x - 6$ has a root or not between $x = 1$ and $x = 2$. if yes, then find the root to five decimal places.

does the given function have a root between the interval $1, 2$? if yes, then find the root. choose the correct answer below.

a. no, the given function has no root between $1, 2$
b. yes, the given function has a root between $1, 2$ and the root is
(type an integer or decimal rounded to five decimal places as needed.)

Question

use the intermediate value theorem to determine whether the function $f(x) = x^3 + 2x - 6$ has a root or not between $x = 1$ and $x = 2$. if yes, then find the root to five decimal places.

does the given function have a root between the interval $1, 2$? if yes, then find the root. choose the correct answer below.

a. no, the given function has no root between $1, 2$
b. yes, the given function has a root between $1, 2$ and the root is
(type an integer or decimal rounded to five decimal places as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Intermediate Value Theorem", "Numerical Root Finding" ], "current_concepts": [ "Intermediate Value Theorem", "Numerical Root Finding" ] } ## Evaluate the function at the interval endpoints $$ \begin{aligned} f(x) &= x^3 + 2x - 6 \ f(1) &= 1^3 + 2(1) - 6 = -3 \ f(2) &= 2^3 + 2(2) - 6 = 6 \end{aligned} $$ ## Apply the Intermediate Value Theorem $$ \begin{aligned} &f(1) = -3 < 0 < 6 = f(2) \ &\text{Since } f(x) \text{ is continuous on } [1, 2], \text{ a root exists in } (1, 2). \end{aligned} $$ ## Find the root to five decimal places $$ \begin{aligned} &x^3 + 2x - 6 = 0 \ &x \approx 1.45553 \end{aligned} $$

A. No, the given function has no root between [1, 2]

B. Yes, the given function has a root between [1, 2] and the root is 1.45553

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Intermediate Value Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the function at the interval endpoints

Apply the Intermediate Value Theorem

Find the root to five decimal places

Answer: