question 5 (5 points)
let $f(x) = x^4 - 4x^3$.
which statement is correct?
$f$ is concave up on $(-\infty, 0)$ and $(2, \infty)$, concave down on $(0, 2)$, and has inflection points at $x = 0$ and $x = 2$.
$f$ is concave down on $(-\infty, 0)$ and $(2, \infty)$, concave up on $(0, 2)$, and has inflection points at $x = 0$ and $x = 2$.
$f$ is concave up on all real numbers and has no inflection points.
$f$ is concave down on all real numbers and has no inflection points.

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Concavity", "Inflection Points" ], "new_concepts": [], "current_concepts": [ "Concavity", "Inflection Points", "Second Derivative Test" ] } ## Find the first and second derivatives Using the **Derivative of Polynomials** knowledge point $$ \begin{aligned} f(x) &= x^4 - 4x^3 \ f'(x) &= 4x^3 - 12x^2 \ f''(x) &= 12x^2 - 24x \end{aligned} $$ ## Find potential inflection points Using the **Inflection Points** knowledge point $$ \begin{aligned} f''(x) &= 0 \ 12x(x - 2) &= 0 \ x = 0 &\quad\text{or}\quad x = 2 \end{aligned} $$ ## Determine concavity intervals Using the **Concavity** knowledge point $$ \begin{aligned} &\text{For } x \in (-\infty, 0): f''(-1) = 12(-1)(-3) = 36 > 0 \implies \text{Concave up} \ &\text{For } x \in (0, 2): f''(1) = 12(1)(-1) = -12 < 0 \implies \text{Concave down} \ &\text{For } x \in (2, \infty): f''(3) = 12(3)(1) = 36 > 0 \implies \text{Concave up} \end{aligned} $$ ## Verify inflection points Using the **Inflection Points** knowledge point $$ \begin{aligned} &\text{Since } f''(x) \text{ changes sign at } x = 0 \text{ and } x = 2, \ &\text{inflection points occur at } x = 0 \text{ and } x = 2. \end{aligned} $$

(A) $f$ is concave up on $(-\infty, 0)$ and $(2, \infty)$, concave down on $(0, 2)$, and has inflection points at $x = 0$ and $x = 2$.

(B) $f$ is concave down on $(-\infty, 0)$ and $(2, \infty)$, concave up on $(0, 2)$, and has inflection points at $x = 0$ and $x = 2$.

(C) $f$ is concave up on all real numbers and has no inflection points.

(D) $f$ is concave down on all real numbers and has no inflection points.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Concavity" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the first and second derivatives

Find potential inflection points

Determine concavity intervals

Verify inflection points

Answer: