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Question
Math
Posted 3 months ago
The derivative of a function ff is given by f(x)=lnxxf^{\prime}(x)=\ln |x| \cdot x.
On which interval is the graph of ff 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 ff is concave up, we need to find the intervals where the second derivative of ff is positive
step 2
The first derivative of ff is given by f(x)=lnxxf'(x) = \ln|x| \cdot x
step 3
The second derivative of ff is f(x)=ddx(lnxx)f''(x) = \frac{d}{dx}(\ln|x| \cdot x). According to the asksia-ll calculation list, f(x)=1xf''(x) = \frac{1}{x}
step 4
The second derivative f(x)=1xf''(x) = \frac{1}{x} is positive when x>0x > 0
step 5
Therefore, the graph of ff is concave up on the interval (0,)(0, \infty)
Answer
The graph of ff is concave up on the interval (0,)(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 1x\frac{1}{x}, which is positive for x>0x > 0.

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