determine where the absolute extrema of (f(x) = ln(x + 72) + \frac{1}{x}) occur on (-28, -1). give exact answers.

the absolute maximum is and it occurs when (x = ).

the absolute minimum is and it occurs when (x = ).

Question

determine where the absolute extrema of (f(x) = ln(x + 72) + \frac{1}{x}) occur on (-28, -1). give exact answers.

the absolute maximum is   and it occurs when (x =  ).

the absolute minimum is   and it occurs when (x =  ).

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Critical Points", "Absolute Extrema on Closed Intervals" ], "new_concepts": [], "current_concepts": [ "Critical Points", "Absolute Extrema on Closed Intervals", "Extreme Value Theorem" ] } ## Find the derivative of the function Using the *Critical Points* knowledge point $$ f'(x) = \frac{1}{x+72} - \frac{1}{x^2} $$ ## Find critical points in the interval Using the *Critical Points* knowledge point $$ \begin{aligned} &\frac{1}{x+72} - \frac{1}{x^2} = 0 \implies x^2 - x - 72 = 0\ &(x-9)(x+8) = 0 \implies x = 9 \text{ or } x = -8 \end{aligned} $$ Since $9 otin [-28, -1]$ and $-8 \in [-28, -1]$, the only relevant critical point is $x = -8$. ## Evaluate the function at the critical point and endpoints Using the *Absolute Extrema on Closed Intervals* knowledge point $$ \begin{aligned} &f(-8) = \ln(-8+72) + \frac{1}{-8} = \ln(64) - \frac{1}{8} = 6\ln(2) - \frac{1}{8} \approx 4.0338\ &f(-28) = \ln(-28+72) + \frac{1}{-28} = \ln(44) - \frac{1}{28} \approx 3.7485\ &f(-1) = \ln(-1+72) + \frac{1}{-1} = \ln(71) - 1 \approx 3.2627 \end{aligned} $$ ## Compare values to determine absolute extrema Using the *Absolute Extrema on Closed Intervals* knowledge point $$ \begin{aligned} &\text{Maximum value: } 6\ln(2) - \frac{1}{8} \quad \text{at } x = -8\ &\text{Minimum value: } \ln(71) - 1 \quad \text{at } x = -1 \end{aligned} $$ Determine where the absolute extrema of $f(x) =\ln (x + 72) + \frac{1}{x}$ occur on $[-28,-1]$. Give exact answers. The absolute maximum is $6\ln(2) - \frac{1}{8}$ and it occurs when $x =$ $-8$. The absolute minimum is $\ln(71) - 1$ and it occurs when $x =$ $-1$. { "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 the derivative of the function

Find critical points in the interval

Evaluate the function at the critical point and endpoints

Compare values to determine absolute extrema

Answer: