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How do you find the inflection points of the graph of the function: #y=1/3x^3#?

Answer 1

Christopher Allers

#(x,y)=(0,0)# is an inflection point.

#y=1/3x^3#

#frac{d^2y}{dx^2}=2x#

At inflection point, second derivative must be zero, which occurs at #x=0#.

#frac{d^2y}{dx^2}|_{x=0}=2(0)=0#

Perform the initial derivative test now.

#frac{dy}{dx}=x^2# changes from positive to zero to positive in a small neighborhood around #x=0#. #x=0# is an inflection point.

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Answer 2

Paisley Johnson

To find the inflection points of the function ( y = \frac{1}{3}x^3 ), we need to find the second derivative and then solve for ( x ) where the second derivative equals zero or is undefined.

First derivative: ( \frac{dy}{dx} = x^2 )

Second derivative: ( \frac{d^2y}{dx^2} = 2x )

Set ( 2x = 0 ) to find where the second derivative is zero, which is at ( x = 0 ).

So, the inflection point is at ( x = 0 ).

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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.