Question

Math

Posted 3 months ago

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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.
```

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Answer from Sia

Posted 3 months ago

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 $x > 0$.

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