How do you find the local extrema for #y = [1 / x]  [1 / (x  1)]#?
graph{1/x  1/(x1) [10.66, 11.84, 3.845, 7.405]}
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To find the local extrema for ( y = \frac{1}{x}  \frac{1}{x  1} ), follow these steps:
 Find the first derivative of ( y ) with respect to ( x ).
 Set the derivative equal to zero and solve for ( x ).
 Check the second derivative at the critical points found in step 2.
 Determine the nature of the local extrema based on the signs of the second derivative.
Let's proceed with these steps:

Find the first derivative ( y' ): [ y' = \frac{1}{x^2} + \frac{1}{(x  1)^2} ]

Set ( y' = 0 ) and solve for ( x ): [ \frac{1}{x^2} + \frac{1}{(x  1)^2} = 0 ]

Solve the equation to find critical points ( x ). [ \frac{1}{(x  1)^2} = \frac{1}{x^2} ] [ (x  1)^2 = x^2 ] [ x^2  2x + 1 = x^2 ] [ 2x + 1 = 0 ] [ x = \frac{1}{2} ]

Check the second derivative at ( x = \frac{1}{2} ): [ y'' = \frac{2}{x^3}  \frac{2}{(x  1)^3} ] Plug in ( x = \frac{1}{2} ): [ y'' = \frac{2}{(\frac{1}{2})^3}  \frac{2}{(\frac{1}{2}  1)^3} = 16 > 0 ]
Since the second derivative is positive at ( x = \frac{1}{2} ), the function has a local minimum at ( x = \frac{1}{2} ).
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When evaluating a onesided 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 onesided 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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