# How do you find points of inflection and determine the intervals of concavity given #y=x^3-3x^2+4#?

concave down on #(-infinity,1) and concave up on (1, infinity)

x=1 is an inflection point.

# y'=3x^2-6x# #y'' = 6x-6 or 6(x-1)# - prepare sign chart with x=1 as a critical point using second derivative.
- to the left of x=1, y'' is negative so concave down. to right of x=1, it is positive so concave up.
- since the concavity changes at x=1, it is an inflection point.

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To find points of inflection and determine intervals of concavity for the function ( y = x^3 - 3x^2 + 4 ), follow these steps:

- Find the second derivative of the function.
- Set the second derivative equal to zero and solve for ( x ). These values are potential points of inflection.
- Test the intervals between these potential points of inflection using the second derivative test to determine concavity.

Let's go through these steps:

- Find the first derivative of the function: ( y' = 3x^2 - 6x ).
- Find the second derivative: ( y'' = 6x - 6 ).
- Set the second derivative equal to zero and solve for ( x ): ( 6x - 6 = 0 ) ( 6x = 6 ) ( x = 1 ).

So, ( x = 1 ) is a potential point of inflection. 4. Test the intervals ( (-\infty, 1) ) and ( (1, +\infty) ) using the second derivative test:

- For ( x = 0 ), ( y'' = -6 < 0 ), indicating concave down.
- For ( x = 2 ), ( y'' = 6 > 0 ), indicating concave up.

Therefore, ( x = 1 ) is a point of inflection, and the interval ( (-\infty, 1) ) is concave down, while the interval ( (1, +\infty) ) is concave up.

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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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