The derivative of a function $f$ is given by $f^{\prime}(x)=\ln |x| \cdot x$.
On which interval is the graph of $f$ concave up? Use a graphing calculator.

Question

The derivative of a function $f$ is given by $f^{\prime}(x)=\ln |x| \cdot x$.
On which interval is the graph of $f$ concave up? Use a graphing calculator.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine where the graph of $f$ is concave up, we need to find the intervals where the second derivative of $f$ is positive. ‖ ‖ step 2 ⋮ The first derivative of $f$ is given by $f'(x) = \ln|x| \cdot x$. ‖ ‖ step 3 ⋮ The second derivative of $f$ is $f''(x) = \frac{d}{dx}(\ln|x| \cdot x)$. According to the asksia-ll calculation list, $f''(x) = \frac{1}{x}$. ‖ ‖ step 4 ⋮ The second derivative $f''(x) = \frac{1}{x}$ is positive when $x > 0$. ‖ ‖ step 5 ⋮ Therefore, the graph of $f$ is concave up on the interval $(0, \infty)$. ‖ ∻Answer∻ ⚹ The graph of $f$ is concave up on the interval $(0, \infty)$. ⚹ ∻Key Concept∻ ⚹Concavity and Second Derivative⚹ ∻Explanation∻ ⚹A graph is concave up where its second derivative is positive. In this case, the second derivative is $\frac{1}{x}$, which is positive for $x > 0$.⚹◊What is the second derivative of the function $f$ given by $f'(x)=\ln |x| \cdot x$?⍭ Generate me a similar question◊

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