# What are the points of inflection of #f(x)=x^3 - 9x^2 + 8x #?

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Points of inflection, also called shifts in concavity, are the places where a function's second derivative changes sign, that is, moves from positive to negative.

Determine the second derivative.

When the second derivative approaches zero, its sign may change.

graph{9x^2+8x^3 [-3, 10, -78.3, 27.1]}

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To find the points of inflection of ( f(x) = x^3 - 9x^2 + 8x ), we need to find the second derivative of the function, ( f''(x) ), and then solve for the values of ( x ) where ( f''(x) = 0 ) or ( f''(x) ) does not exist.

First, find the first derivative: [ f'(x) = 3x^2 - 18x + 8 ]

Then, find the second derivative: [ f''(x) = 6x - 18 ]

Next, set ( f''(x) ) equal to zero and solve for ( x ): [ 6x - 18 = 0 ] [ 6x = 18 ] [ x = 3 ]

So, ( x = 3 ) is a potential point of inflection.

Now, we need to test the concavity of the function around this point. We can do this by choosing test points on either side of ( x = 3 ) and plugging them into ( f''(x) ).

For ( x < 3 ), choose ( x = 2 ): [ f''(2) = 6(2) - 18 = -6 ]

For ( x > 3 ), choose ( x = 4 ): [ f''(4) = 6(4) - 18 = 6 ]

Since the sign of ( f''(x) ) changes from negative to positive at ( x = 3 ), ( x = 3 ) is a point of inflection.

So, the point of inflection of ( f(x) = x^3 - 9x^2 + 8x ) is ( (3, f(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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