How do you find the monotonicity, extrema, concavity, and inflection points of #f(x)=lnx/sqrt(x)#?
graph{lnx/sqrtx [0,5, 1000, 2.88, 2]}
First, we note that the function:
We are able to examine the behavior at the domain's boundaries:
Let's compute the function's first and second derivatives:
We can now observe that:
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To find the monotonicity, extrema, concavity, and inflection points of ( f(x) = \frac{\ln(x)}{\sqrt{x}} ), follow these steps:

Monotonicity: Determine where the derivative ( f'(x) ) is positive or negative to find where ( f(x) ) is increasing or decreasing.

Extrema: Find critical points by setting ( f'(x) = 0 ) and then use the first or second derivative test to determine whether these points correspond to local extrema.

Concavity: Find the second derivative ( f''(x) ) and determine where it is positive or negative to identify intervals of concavity.

Inflection Points: Locate points where the concavity changes sign, indicating potential inflection points. These occur where ( f''(x) = 0 ) or where ( f''(x) ) is undefined.
Let's proceed with the calculations:

Monotonicity: Calculate the first derivative ( f'(x) ): [ f'(x) = \frac{1  2\ln(x)}{2x^{3/2}} ]
Set ( f'(x) = 0 ) to find critical points: [ 1  2\ln(x) = 0 ] Solving for ( x ), we get ( x = e^{1/2} ).
Test intervals around the critical point to determine monotonicity.

Extrema: Use the first derivative test or second derivative test to determine if ( x = e^{1/2} ) corresponds to an extremum.

Concavity: Calculate the second derivative ( f''(x) ): [ f''(x) = \frac{3 + 4\ln(x)}{4x^{5/2}} ]
Determine intervals where ( f''(x) > 0 ) or ( f''(x) < 0 ) to identify concave up or concave down intervals.

Inflection Points: Locate points where ( f''(x) = 0 ) or where ( f''(x) ) is undefined to find potential inflection points.
After performing these calculations, you can analyze the behavior of the function ( f(x) ) with respect to monotonicity, extrema, concavity, and inflection 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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