# How do you use the first and second derivatives to sketch #y = x - ln |x|#?

so this critical point is a local minimum. It has no inflection points and is concave up in its domain.

We can also note that:

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To sketch the graph of ( y = x - \ln |x| ) using the first and second derivatives:

- Find the first derivative of ( y ) with respect to ( x ).
- Find critical points by setting the first derivative equal to zero and solving for ( x ).
- Determine the intervals of increase and decrease by analyzing the sign of the first derivative.
- Find the second derivative of ( y ) with respect to ( x ).
- Determine the intervals of concavity by analyzing the sign of the second derivative.
- Find any inflection points by setting the second derivative equal to zero and solving for ( x ).
- Sketch the graph using the information obtained from steps 3, 5, and 6, along with any additional points of interest such as intercepts or asymptotes.

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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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- How do you find all local maximum and minimum points using the second derivative test given #y=x^2-x#?

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