a) for what values of $x$ does the function $f$ have a local maximum on $(0, 8)$?

b) for what values of $x$ does the function $f$ have a local minimum on $(0, 8)$?

c) find the absolute maximum for the function $f$ on the interval $(0, 8)$.

d) find the absolute minimum for the function $f$ on the interval $(0, 8)$.

Question

a) for what values of $x$ does the function $f$ have a local maximum on $(0, 8)$?

b) for what values of $x$ does the function $f$ have a local minimum on $(0, 8)$?

c) find the absolute maximum for the function $f$ on the interval $(0, 8)$.

d) find the absolute minimum for the function $f$ on the interval $(0, 8)$.

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Local Extrema", "Absolute Extrema" ], "current_concepts": [ "Local Extrema", "Absolute Extrema" ] } ## Identify local extrema on the open interval (0, 8) $$ \begin{aligned} &\text{Local maximum: } x = 3 \text{ because } f(3) = 6 \ge f(x) \text{ for all } x \text{ in a neighborhood of } 3.\ &\text{Local minimum: } x = 2 \text{ and } x = 5.\ &\quad \text{At } x = 2, f(2) = 4 \le f(x) \text{ for all } x \text{ in a neighborhood of } 2.\ &\quad \text{At } x = 5, f(5) = 2 \le f(x) \text{ for all } x \text{ in a neighborhood of } 5. \end{aligned} $$ ## Identify absolute maximum on the open interval (0, 8) $$ \begin{aligned} &\text{The supremum of } f(x) \text{ on } (0, 8) \text{ is } 7 \text{ as } x \to 0^+.\ &\text{Since } 0 otin (0, 8) \text{ and } f(x) < 7 \text{ for all } x \in (0, 8), \text{ the absolute maximum does not exist.} \end{aligned} $$ ## Identify absolute minimum on the open interval (0, 8) $$ \begin{aligned} &\text{The lowest value attained by } f(x) \text{ on } (0, 8) \text{ is } 2 \text{ at } x = 5.\ &\text{Absolute minimum value is } 2 \text{ (attained at } x = 5\text{).} \end{aligned} $$ a) $x = 3$ b) $x = 2, 5$ c) Does not exist d) $2$ (at $x = 5$) { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Local Extrema" ] }

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Explanation:

Identify local extrema on the open interval (0, 8)

Identify absolute maximum on the open interval (0, 8)

Identify absolute minimum on the open interval (0, 8)

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