How do you find increasing, decreasing, inflection points, minimum and maximum for the graph #f(x) = ln(x)/(8sqrtx)#?
See the explanation.
The first and second derivatives must be found:
The first derivative's zeros are stationary points:
If the second derivative changes sign at those points and the function is continuous, the points of inflection are zeros of the second derivative:
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To find increasing and decreasing intervals, inflection points, minimum, and maximum for ( f(x) = \frac{\ln(x)}{8\sqrt{x}} ):
 Find the first derivative: ( f'(x) = \frac{1}{8x\sqrt{x}}  \frac{\ln(x)}{16x^{\frac{3}{2}}} ).
 Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.
 Determine the intervals where ( f'(x) > 0 ) for increasing intervals and where ( f'(x) < 0 ) for decreasing intervals.
 Find the second derivative: ( f''(x) ).
 Determine the sign of ( f''(x) ) in the intervals between critical points to find inflection points.
 Check the behavior of the function at the critical points and endpoints to find minimum and maximum points.
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To find increasing, decreasing, inflection points, minimum, and maximum for the graph of ( f(x) = \frac{\ln(x)}{8\sqrt{x}} ):

Increasing and Decreasing Intervals:
 ( f(x) ) is increasing when its derivative is positive.
 ( f(x) ) is decreasing when its derivative is negative.

First Derivative:
 Find the first derivative of ( f(x) ) using the quotient rule.
 ( f'(x) = \frac{d}{dx} \left( \frac{\ln(x)}{8\sqrt{x}} \right) )

Second Derivative:
 To find inflection points, we need the second derivative.
 ( f''(x) = \frac{d^2}{dx^2} \left( \frac{\ln(x)}{8\sqrt{x}} \right) )

Critical Points:
 Critical points occur where ( f'(x) = 0 ) or ( f'(x) ) is undefined.
 Solve ( f'(x) = 0 ) to find critical points.

Minimum and Maximum Points:
 Use the first derivative test or the second derivative test to determine minimum and maximum points.
 If ( f''(x) > 0 ), it's a minimum point.
 If ( f''(x) < 0 ), it's a maximum point.

Procedure Summary:
 Find ( f'(x) ) and ( f''(x) ).
 Determine where ( f'(x) = 0 ) and any points where ( f'(x) ) is undefined.
 Test intervals between critical points using the first or second derivative test.
 Identify increasing/decreasing intervals and minimum/maximum points.
Following this procedure will help in analyzing the function ( f(x) = \frac{\ln(x)}{8\sqrt{x}} ) for its increasing/decreasing behavior, inflection points, and minimum/maximum points.
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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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