consider the function $f(x) = 4 - 2x^2$, $-4 \le x \le 2$.

the absolute maximum value is
and this occurs at $x =$

the absolute minimum value is
and this occurs at $x =$

Question

consider the function $f(x) = 4 - 2x^2$, $-4 \le x \le 2$.

the absolute maximum value is
and this occurs at $x =$

the absolute minimum value is
and this occurs at $x =$

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", "Local Extrema Classification" ], "new_concepts": [ "Absolute Extrema on Closed Intervals" ], "current_concepts": [ "Critical Points", "Local Extrema Classification", "Absolute Extrema on Closed Intervals" ] } ## Find critical points in the interval $$ \begin{aligned} f(x) &= 4 - 2x^2, \quad x \in [-4, 2] \ f'(x) &= -4x \ f'(x) &= 0 \implies x = 0 \in [-4, 2] \end{aligned} $$ ## Evaluate the function at critical points and endpoints $$ \begin{aligned} f(0) &= 4 - 2(0)^2 = 4 \ f(-4) &= 4 - 2(-4)^2 = 4 - 32 = -28 \ f(2) &= 4 - 2(2)^2 = 4 - 8 = -4 \end{aligned} $$ ## Identify absolute maximum and minimum values $$ \begin{aligned} \text{Absolute Maximum Value} &= 4 \quad \text{at } x = 0 \ \text{Absolute Minimum Value} &= -28 \quad \text{at } x = -4 \end{aligned} $$ Consider the function $f(x) = 4 - 2x^2$, $-4 \le x \le 2$. The absolute maximum value is 4 and this occurs at $x =$ 0 The absolute minimum value is -28 and this occurs at $x =$ -4 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Absolute Extrema on Closed Intervals" ] }

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QUESTION IMAGE

Question

Explanation:

Find critical points in the interval

Evaluate the function at critical points and endpoints

Identify absolute maximum and minimum values

Answer: