# What are the points of inflection, if any, of #f(x) =(x+4)/(x-2)^2#?

The second derivative must vanish in order for the function to have an inflection point.

Determine the second derivative by:

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To find the points of inflection, we first need to find the second derivative of the function and then determine where it changes sign.

The first derivative of ( f(x) ) is ( f'(x) = \frac{-2x^2 - 4x + 16}{(x - 2)^3} ).

The second derivative is ( f''(x) = \frac{2x^3 - 4x^2 - 32x + 48}{(x - 2)^4} ).

To find the points of inflection, we need to find where ( f''(x) = 0 ) or does not exist, and then check the sign of ( f''(x) ) around those points.

Solving ( f''(x) = 0 ) gives us ( x = 2 ).

At ( x = 2 ), the second derivative changes sign from negative to positive, indicating a point of inflection.

Therefore, the point of inflection of the function ( f(x) = \frac{x+4}{(x-2)^2} ) is at ( x = 2 ).

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