What are the points of inflection, if any, of #f(x) =x^3 - 12x^2 #?

Answer 1

#x=4#

Points of inflection are where #f''(x)=0# or when #f''(x)# instantaneously changes from positive to negative or vice versa.
#f(x) = x^3-12x^2# is a polynomial -- it is a smooth curve so we don't have to worry about instantaneous jumps (e.g. asymptotes).
All we have to do is differentiate twice and set that equal to #0#.
#f'(x)=3x^2-24x#
#f''(x) = 6x-24#
Now set #f''(x)# equal to #0#.
#6x-24=0# #6x=24# #x=4#
This means that there is one point of inflection on the graph of #f(x)# and it is #x=4#.

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Answer 2

To find the points of inflection, we need to determine where the concavity changes. We find the second derivative of ( f(x) = x^3 - 12x^2 ) and solve for where it equals zero.

First derivative: ( f'(x) = 3x^2 - 24x ) Second derivative: ( f''(x) = 6x - 24 )

Setting ( f''(x) ) equal to zero and solving for ( x ): ( 6x - 24 = 0 ) ( 6x = 24 ) ( x = 4 )

So, the point of inflection is at ( x = 4 ).

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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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