How do you find all points of inflection given #y=((x3)/(x+1))^2#?
The only inflection point is
Calculate the second derivative:
So, the only candidate inflection point is:
graph{((x3)/(x+1))^2 [21, 19, 10.48, 9.52]}
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To find the points of inflection for ( y = \left(\frac{x3}{x+1}\right)^2 ), follow these steps:
 Find the second derivative of the function.
 Set the second derivative equal to zero and solve for ( x ).
 Determine the ( y )coordinate of each point of inflection by substituting the ( x )values into the original function.
Let's go through these steps:

First derivative: ( y' = 2\left(\frac{x3}{x+1}\right) \left(\frac{(x+1)  (x3)}{(x+1)^2}\right) )

Second derivative: ( y'' = 2\left(\frac{x3}{x+1}\right) \left(\frac{(x+1)  (x3)}{(x+1)^2}\right) + 2\left(\frac{x3}{x+1}\right)^2 \left(\frac{1}{(x+1)^2}\right) )

Set ( y'' ) equal to zero and solve for ( x ) to find critical points.

Once you find the critical points, determine if they are points of inflection by evaluating the concavity of the function around these points.

Find the ( y )coordinates of the points of inflection by substituting the ( x )values into the original function.
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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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