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

Answer 1

#f(x)# has an inflection point for #x=-16#.

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

Determine the second derivative by:

#f(x) = frac (x+4) ((x-2)^2)#
#f'(x) = frac ((x-2)^2-2(x-2)(x+4)) ((x-2)^4) = frac ((x-2)-2(x+4)) ((x-2)^3) = frac (x-2-2x-8) ((x-2)^3)= -frac (x+10) ((x-2)^3)#
#f''(x) = frac (-(x-2)^3+3(x+10)(x-2)^2) ((x-2)^6) = frac (-(x-2)+3(x+10)) ((x-2)^4) = frac (-x+2+3x+30) ((x-2)^4) = 2 frac (x+16) ((x-2)^4)#
So #f''(x) = 0# for #x=-16#.
To be sure this is effectively an inflection point we have to check that #f''(x)# changes sign around #x=-16#. We can see that the denominator is always positive, and the numerator is of first grade in x, so it definitely changes sign around the root.
We can conclude that #x=-16# is an inflection point.
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Answer 2

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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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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