# How do you determine whether the function #y=ln x# is concave up or concave down and its intervals?

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To determine the concavity of the function ( y = \ln(x) ) and its intervals:

- Find the second derivative ( y'' ) of ( y = \ln(x) ).
- Set ( y'' ) equal to zero to find the points of inflection.
- Test the intervals between these points of inflection with the second derivative test:
- If ( y'' > 0 ) in an interval, the function is concave up on that interval.
- If ( y'' < 0 ) in an interval, the function is concave down on that interval.

For ( y = \ln(x) ):

- First derivative ( y' = \frac{1}{x} ).
- Second derivative ( y'' = -\frac{1}{x^2} ).

Since ( y'' = -\frac{1}{x^2} ):

- ( y'' > 0 ) for ( x > 0 ), indicating ( y = \ln(x) ) is concave up on ( (0, \infty) ).
- ( y'' < 0 ) for ( x < 0 ), but the natural logarithm function is not defined for ( x \leq 0 ). Thus, there are no intervals where ( y = \ln(x) ) is concave down.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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