consider the function $f(x) = 5 - 5x^2$, $-4 \le x \le 2$.

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

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

Question

consider the function $f(x) = 5 - 5x^2$, $-4 \le x \le 2$.

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": [ "Absolute Extrema on Closed Intervals", "Critical Points" ], "new_concepts": [], "current_concepts": [ "Absolute Extrema on Closed Intervals", "Critical Points" ] } ## Find critical points of the function $$ \begin{aligned} f(x) &= 5 - 5x^2, \quad -4 \le x \le 2 \ f'(x) &= -10x \ -10x &= 0 \implies x = 0 \end{aligned} $$ ## Evaluate the function at critical points and endpoints $$ \begin{aligned} f(0) &= 5 - 5(0)^2 = 5 \ f(-4) &= 5 - 5(-4)^2 = 5 - 5(16) = -75 \ f(2) &= 5 - 5(2)^2 = 5 - 5(4) = -15 \end{aligned} $$ ## Identify absolute extrema $$ \begin{aligned} \text{Absolute Maximum Value} &= 5 \quad \text{at } x = 0 \ \text{Absolute Minimum Value} &= -75 \quad \text{at } x = -4 \end{aligned} $$ Consider the function $f(x) = 5 - 5x^2$, $-4 \le x \le 2$. The absolute maximum value is 5 and this occurs at $x =$ 0 The absolute minimum value is -75 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 of the function

Evaluate the function at critical points and endpoints

Identify absolute extrema

Answer: