the function $f(x) = 2x^3 - 30x^2 + 96x + 4$ has one local minimum and one local maximum.
this function has a local minimum at $x$ equals
with value
and a local maximum at $x$ equals
with value

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Critical Points", "Local Extrema" ], "new_concepts": [], "current_concepts": [ "Critical Points", "Local Extrema", "First Derivative Test" ] } ## Find the first derivative Using the *Critical Points* knowledge point $$ f'(x) = 6x^2 - 60x + 96 $$ ## Solve for critical points Using the *Critical Points* knowledge point $$ \begin{aligned} 6(x^2 - 10x + 16) &= 0 \ 6(x - 2)(x - 8) &= 0 \ x = 2, \quad x = 8 \end{aligned} $$ ## Classify local extrema Using the *Local Extrema* knowledge point $$ \begin{aligned} f''(x) &= 12x - 60 \ f''(2) &= 12(2) - 60 = -36 < 0 \implies \text{local maximum at } x = 2 \ f''(8) &= 12(8) - 60 = 36 > 0 \implies \text{local minimum at } x = 8 \end{aligned} $$ ## Calculate local extrema values Using the *Local Extrema* knowledge point $$ \begin{aligned} f(2) &= 2(2)^3 - 30(2)^2 + 96(2) + 4 = 16 - 120 + 192 + 4 = 92 \ f(8) &= 2(8)^3 - 30(8)^2 + 96(8) + 4 = 1024 - 1920 + 768 + 4 = -124 \end{aligned} $$ The function $f(x) = 2x^3 - 30x^2 + 96x + 4$ has one local minimum and one local maximum. This function has a local minimum at $x$ equals 8 with value -124 and a local maximum at $x$ equals 2 with value 92. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Local Extrema" ] }

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Explanation:

Find the first derivative

Solve for critical points

Classify local extrema

Calculate local extrema values

Answer: