How do you find the intervals of increasing and decreasing using the first derivative given #y=x+4/x#?
The intervals of increasing are
The intervals of decreasing are
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We figure out the initial derivative.
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To find the intervals of increasing and decreasing using the first derivative ( \frac{dy}{dx} ) given ( y = x + \frac{4}{x} ):
 Find the first derivative ( \frac{dy}{dx} ) of the given function ( y = x + \frac{4}{x} ).
 Set the first derivative equal to zero and solve for ( x ) to find critical points.
 Determine the intervals where the first derivative is positive to identify where the function is increasing.
 Determine the intervals where the first derivative is negative to identify where the function is decreasing.
Let's proceed with the steps:

Find the first derivative: [ \frac{dy}{dx} = 1  \frac{4}{x^2} ]

Set ( \frac{dy}{dx} ) equal to zero: [ 1  \frac{4}{x^2} = 0 ] [ 1 = \frac{4}{x^2} ] [ x^2 = 4 ] [ x = \pm 2 ]
So, the critical points are ( x = 2 ) and ( x = 2 ).
 Test intervals to find where the first derivative is positive or negative:
 For ( x < 2 ): Choose ( x = 3 ), ( \frac{dy}{dx} = 1  \frac{4}{(3)^2} = \frac{5}{9} < 0 )
 For ( 2 < x < 2 ): Choose ( x = 0 ), ( \frac{dy}{dx} = 1  \frac{4}{0^2} ) is undefined, so we choose a value slightly above and below 0:
 For ( x = 1 ), ( \frac{dy}{dx} = 1  \frac{4}{1^2} = 3 < 0 )
 For ( x = 1 ), ( \frac{dy}{dx} = 1  \frac{4}{(1)^2} = 1 > 0 )
 For ( x > 2 ): Choose ( x = 3 ), ( \frac{dy}{dx} = 1  \frac{4}{3^2} = \frac{5}{9} > 0 )
Based on these tests:
 The function ( y = x + \frac{4}{x} ) is decreasing for ( x < 2 ) and ( 2 < x < 2 ).
 The function ( y = x + \frac{4}{x} ) is increasing for ( x > 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.
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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