Questions: f(x)=(x+12) * ln(x+4) for x>-4. f(x) is concave up on the interval ( , ∞ )

Transcript text: webwork.as.uky.edu webwork / ma123f24 / hw10_concavity_and_curve_sketching / 6 HW10 Concavity and Curve Sketching: Pr Previous Problem Problem List Next Problem (1 point) Let \[ f(x)=(x+12) \cdot \ln (x+4) \] for $x>-4$. $f(x)$ is concave up on the interval ( $\square$ , $\infty$ ) Preview My Answers Submit Answers Show me another You have attempted this problem 0 times. You have unlimited attempts remaining. Show Past Answers Page generated at 11/07/2024 at 04:02pm EST WeBWorK (C) 1996-2019| theme: math4 | ww_version: 2.15|pg_version 2.15| The WeBWorK Project

Solution

Solution Steps

To determine where the function $ f(x) = (x+12) \cdot \ln(x+4) $ is concave up, we need to find the second derivative of the function and analyze its sign. The function is concave up where the second derivative is positive. First, compute the first derivative using the product rule, then find the second derivative. Finally, solve the inequality where the second derivative is greater than zero to find the interval of concavity.

Step 1: Find the First Derivative

To find the concavity of the function $ f(x) = (x + 12) \cdot \ln(x + 4) $, we first compute the first derivative $ f'(x) $: \[ f'(x) = \ln(x + 4) + \frac{x + 12}{x + 4} \]

Step 2: Find the Second Derivative

Next, we compute the second derivative $ f''(x) $: \[ f''(x) = \frac{2}{x + 4} - \frac{x + 12}{(x + 4)^2} \]

Step 3: Determine the Interval of Concavity

To find where the function is concave up, we solve the inequality $ f''(x) > 0 $. The solution indicates that the function is concave up for: \[ x > 4 \]