How do you find the points of inflection for #f(x)=x sqrt(x+1)#?
There are no inflection points for this
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To find the points of inflection for ( f(x) = x \sqrt{x+1} ):

Find the second derivative of ( f(x) ): [ f'(x) = \frac{d}{dx}(x \sqrt{x+1}) ] [ = \sqrt{x+1} + x \cdot \frac{d}{dx}(\sqrt{x+1}) ] [ = \sqrt{x+1} + x \cdot \frac{1}{2 \sqrt{x+1}} \cdot \frac{d}{dx}(x+1) ] [ = \sqrt{x+1} + \frac{x}{2 \sqrt{x+1}} \cdot 1 ] [ = \sqrt{x+1} + \frac{x}{2 \sqrt{x+1}} ]

Find the second derivative: [ f''(x) = \frac{d}{dx}(\sqrt{x+1} + \frac{x}{2 \sqrt{x+1}}) ] [ = \frac{1}{2 \sqrt{x+1}} + \frac{1}{2 \sqrt{x+1}}  \frac{x}{4(x+1)^{3/2}} ] [ = \frac{1}{\sqrt{x+1}}  \frac{x}{4(x+1)^{3/2}} ]

Set ( f''(x) = 0 ) to find potential points of inflection: [ \frac{1}{\sqrt{x+1}}  \frac{x}{4(x+1)^{3/2}} = 0 ]
Solving this equation will give you the ( x )values of potential points of inflection.

Check the concavity of the function: Evaluate ( f''(x) ) at the points found in step 3 to determine whether the concavity changes at those points.

Confirm points of inflection: Confirm that the concavity changes at each point found in step 3 to identify the points of inflection.
This process will give you the points of inflection for the function ( f(x) = x \sqrt{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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