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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)
The graph of ff is concave up on the interval (0,)(0, \infty).
Key Concept
Concavity and Second Derivative
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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