# What are the points of inflection of #f(x)=2 / (x^2 - 9) #?

There are no points of inflection for the graph of this function.

A point on the graph where the concavoty changes is called an inflection point.

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To find the points of inflection of ( f(x) = \frac{2}{x^2 - 9} ), we need to first find the second derivative of the function. Then, we set the second derivative equal to zero and solve for the values of ( x ) to determine the points of inflection.

First, let's find the first derivative of ( f(x) ): [ f'(x) = \frac{d}{dx} \left( \frac{2}{x^2 - 9} \right) ]

[ f'(x) = \frac{-4x}{(x^2 - 9)^2} ]

Now, let's find the second derivative: [ f''(x) = \frac{d}{dx} \left( \frac{-4x}{(x^2 - 9)^2} \right) ]

[ f''(x) = \frac{-4(x^2 - 9)^2 - 2(-4x)(2x(x^2 - 9))}{(x^2 - 9)^4} ]

[ f''(x) = \frac{-4(x^4 - 18x^2 + 81) + 16x^2(x^2 - 9)}{(x^2 - 9)^4} ]

[ f''(x) = \frac{-4x^4 + 72x^2 - 324 + 16x^4 - 144x^2}{(x^2 - 9)^4} ]

[ f''(x) = \frac{12x^4 - 72x^2 - 324}{(x^2 - 9)^4} ]

To find the points of inflection, we set the second derivative equal to zero and solve for ( x ): [ 12x^4 - 72x^2 - 324 = 0 ]

[ 3x^4 - 18x^2 - 81 = 0 ]

[ (x^2 - 9)(3x^2 + 9) = 0 ]

[ x^2 - 9 = 0 \quad \text{(Ignoring the trivial solution)} ]

[ x^2 = 9 ]

[ x = \pm 3 ]

Therefore, the points of inflection of ( f(x) = \frac{2}{x^2 - 9} ) are ( x = -3 ) and ( x = 3 ).

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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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