the function $f(x) = -2x^3 + 36x^2 - 162x + 7$ has one local minimum and one local maximum.
this function has a local minimum at $x =$
with value
and a local maximum at $x =$
with value

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Critical Points", "First Derivative Test", "Local Extrema Classification" ], "new_concepts": [], "current_concepts": [ "Critical Points", "First Derivative Test", "Local Extrema Classification" ] } ## Find the first derivative and critical points $$ \begin{aligned} f(x) &= -2x^3 + 36x^2 - 162x + 7 \ f'(x) &= -6x^2 + 72x - 162 \ -6(x^2 - 12x + 27) &= 0 \ -6(x - 3)(x - 9) &= 0 \ x &= 3, \quad x = 9 \end{aligned} $$ ## Classify the local extrema $$ \begin{aligned} f''(x) &= -12x + 72 \ f''(3) &= -12(3) + 72 = 36 > 0 \implies \text{Local minimum at } x = 3 \ f''(9) &= -12(9) + 72 = -36 < 0 \implies \text{Local maximum at } x = 9 \end{aligned} $$ ## Calculate the local minimum and maximum values $$ \begin{aligned} f(3) &= -2(3)^3 + 36(3)^2 - 162(3) + 7 \ &= -54 + 324 - 486 + 7 \ &= -209 \ f(9) &= -2(9)^3 + 36(9)^2 - 162(9) + 7 \ &= -1458 + 2916 - 1458 + 7 \ &= 7 \end{aligned} $$ The function $f(x) = -2x^3 + 36x^2 - 162x + 7$ has one local minimum and one local maximum. This function has a local minimum at $x =$ 3 with value -209 and a local maximum at $x =$ 9 with value 7 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Local Extrema Classification" ] }

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Question

Explanation:

Find the first derivative and critical points

Classify the local extrema

Calculate the local minimum and maximum values

Answer: