How do you find all local maximum and minimum points using the second derivative test given #y=x+1/x#?
Please see the explanation section below
Find the second derivative
Apply the test
Again, I assume the requested form for the answer is
Additional note
We have finished answering the question, but the answer may look strange to students. (The relative minimum is greater than the relative maximum.)
graph{x+1/x [10.38, 12.12, 7.335, 3.915]}
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To find all local maximum and minimum points using the second derivative test for the function ( y = x + \frac{1}{x} ):

Find the first derivative of the function: [ y' = 1  \frac{1}{x^2} ]

Find the critical points by setting the first derivative equal to zero and solving for ( x ): [ 1  \frac{1}{x^2} = 0 ] [ 1 = \frac{1}{x^2} ] [ x^2 = 1 ] [ x = \pm 1 ]

Determine the second derivative of the function: [ y'' = \frac{2}{x^3} ]

Evaluate the second derivative at the critical points: [ y''(1) = 2 ] [ y''(1) = 2 ]

Interpret the results:
 If ( y''(c) > 0 ), then the function has a local minimum at ( x = c ).
 If ( y''(c) < 0 ), then the function has a local maximum at ( x = c ).

Therefore, at ( x = 1 ), ( y''(1) > 0 ), so there is a local minimum at ( x = 1 ). At ( x = 1 ), ( y''(1) < 0 ), so there is a local maximum at ( x = 1 ).
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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.
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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