# What are the points of inflection, if any, of #f(x) = -3x^4-2 x^3 +9 #?

To locate the inflection points, we must locate the roots of

therefore from

Consequently, the roots are

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To find the points of inflection, you need to calculate the second derivative of the function and then solve for where the second derivative equals zero or does not exist. Let's find the second derivative of ( f(x) ):

[ f(x) = -3x^4 - 2x^3 + 9 ]

First derivative:

[ f'(x) = -12x^3 - 6x^2 ]

Second derivative:

[ f''(x) = -36x^2 - 12x ]

Now, set ( f''(x) ) equal to zero and solve for ( x ):

[ -36x^2 - 12x = 0 ]

[ -12x(3x + 1) = 0 ]

[ x = 0, \frac{-1}{3} ]

So, the points of inflection are at ( x = 0 ) and ( x = \frac{-1}{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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