find the critical numbers of the function $f(x) = 12x^5 + 75x^4 - 20x^3 - 4$ and classify them. round your answers to three decimal places.

all critical numbers must be entered and categorized before submitting your answer, otherwise your answer will be marked as incorrect.

$x =$ is a select an answer
$x =$ is a select an answer
$x =$ is a select an answer

Question

find the critical numbers of the function $f(x) = 12x^5 + 75x^4 - 20x^3 - 4$ and classify them. round your answers to three decimal places.

all critical numbers must be entered and categorized before submitting your answer, otherwise your answer will be marked as incorrect.

$x =$   is a select an answer
$x =$   is a select an answer
$x =$   is a select an answer

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Critical Points" ], "new_concepts": [ "First Derivative Test", "Local Extrema Classification" ], "current_concepts": [ "Critical Points", "First Derivative Test", "Local Extrema Classification" ] } ## Find the derivative of the function $$ \begin{aligned} f(x) &= 12x^5 + 75x^4 - 20x^3 - 4 \ f'(x) &= 60x^4 + 300x^3 - 60x^2 \ f'(x) &= 60x^2(x^2 + 5x - 1) \end{aligned} $$ ## Solve for critical numbers where the derivative is zero $$ \begin{aligned} 60x^2(x^2 + 5x - 1) &= 0 \ x_1 &= 0 \ x^2 + 5x - 1 &= 0 \implies x = \frac{-5 \pm \sqrt{25 - 4(1)(-1)}}{2} = \frac{-5 \pm \sqrt{29}}{2} \ x_2 &= \frac{-5 - \sqrt{29}}{2} \approx -5.193 \ x_3 &= \frac{-5 + \sqrt{29}}{2} \approx 0.193 \end{aligned} $$ ## Classify the critical numbers using the first derivative test $$ \begin{aligned} &\text{For } x < -5.193: f'(x) > 0 \ &\text{For } -5.193 < x < 0: f'(x) < 0 \implies x \approx -5.193 \text{ is a local maximum} \ &\text{For } 0 < x < 0.193: f'(x) < 0 \implies x = 0 \text{ is neither a local minimum nor a local maximum} \ &\text{For } x > 0.193: f'(x) > 0 \implies x \approx 0.193 \text{ is a local minimum} \end{aligned} $$ $x = -5.193$ is a local maximum $x = 0$ is neither a local minimum nor a local maximum $x = 0.193$ is a local minimum { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Critical Points" ] }

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QUESTION IMAGE

Question

Explanation:

Find the derivative of the function

Solve for critical numbers where the derivative is zero

Classify the critical numbers using the first derivative test

Answer: