Home > Calculus > Determining Points of Inflection for a Function

How do you find the inflection points of #f(x)=x^5-30x^3#?

Answer 1

0 and -3

#df(x)=5x^4-90x^2# #d(df(x))=20x^3-180x# #20x^3-180x=0# #x=0; x=3; x=-3# #g(x)=d(d(df(x)))=60x^2-180# #g(0)=-180; g(3)=0; g(-3)=1080# 3 is not infection point.

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Answer 2

Christopher Agresta

To find the inflection points of ( f(x) = x^5 - 30x^3 ), we need to determine where the concavity changes.

Find the second derivative of ( f(x) ): ( f''(x) = \frac{d^2}{dx^2}(x^5 - 30x^3) ).
Set ( f''(x) = 0 ) and solve for ( x ) to find the points where the concavity may change.
Once you have the values of ( x ) where ( f''(x) = 0 ) or ( f''(x) ) is undefined, plug those values into the first derivative ( f'(x) ) to determine the concavity at those points.
Confirm the concavity changes by checking the signs of ( f''(x) ) around these points.
The points where the concavity changes are the inflection points.

So, in this case, you would find ( f''(x) ) and then solve ( f''(x) = 0 ) to identify the potential inflection points. After that, you would use the first derivative test to confirm the concavity changes and determine the inflection points.

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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.