How do you find the points of inflection for #f(x)=x sqrt(x+1)#?

Answer 1

There are no inflection points for this #f#.

An inflection point is a point of the graph of #f# at which the concavity changes. It is a point on the graph at which the sign of #f''# changes.
#f(x) = xsqrt(x+1)# has domain #[-1,oo)#
#f''(x) = (3x+4)/(4(x+1)^(3/2))# is positive on #(-1,oo)# so there are no inflection points for #f#.
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Answer 2

To find the points of inflection for ( f(x) = x \sqrt{x+1} ):

  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}} ]

  2. 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}} ]

  3. 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.

  4. 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.

  5. 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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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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