find and classify the absolute extrema of the function $g(x) = x^2 - 49$ on the domain $d = (-7,7)$. if the function doesnt have an absolute extremum, write none for your answer.

absolute minimum value:
absolute maximum value:

Question

find and classify the absolute extrema of the function $g(x) = x^2 - 49$ on the domain $d = (-7,7)$. if the function doesnt have an absolute extremum, write none for your answer.

absolute minimum value:
absolute maximum value:

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Absolute Extrema on Closed Intervals" ], "new_concepts": [ "Absolute Extrema on Open Intervals" ], "current_concepts": [ "Absolute Extrema on Closed Intervals", "Absolute Extrema on Open Intervals" ] } ## Find the critical points of the function $$ \begin{aligned} g(x) &= x^2 - 49 \ g'(x) &= 2x \ 2x &= 0 \implies x = 0 \end{aligned} $$ ## Evaluate the function at the critical point $$ \begin{aligned} 0 &\in (-7, 7) \ g(0) &= 0^2 - 49 = -49 \end{aligned} $$ ## Analyze the behavior near the boundary of the open domain $$ \begin{aligned} &\lim_{x \to -7^+} g(x) = (-7)^2 - 49 = 0 \ &\lim_{x \to 7^-} g(x) = (7)^2 - 49 = 0 \ &\forall x \in (-7, 7), \quad g(x) \ge -49 \ &\text{Absolute minimum value: } -49 \ &\text{Absolute maximum value: None} \end{aligned} $$ Absolute minimum value: -49 Absolute maximum value: None { "elements": [ { "type": "functiongraph", "params": [ { "js": "x*x - 49", "latex": "x^2 - 49" }, -6.99, 6.99 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "g(x)", "withLabel": true } }, { "type": "point", "params": [ [0, -49] ], "properties": { "name": "Minimum (0, -49)", "size": 4, "color": "#F2557F", "strokeColor": "#F2557F", "fillColor": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [-7, 0] ], "properties": { "name": "(-7, 0)", "size": 4, "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "white", "withLabel": true } }, { "type": "point", "params": [ [7, 0] ], "properties": { "name": "(7, 0)", "size": 4, "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "white", "withLabel": true } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Absolute Extrema on Open Intervals" ] }

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Question

Explanation:

Find the critical points of the function

Evaluate the function at the critical point

Analyze the behavior near the boundary of the open domain

Answer: