Home > Calculus > Determining Points of Inflection for a Function

What are the points of inflection, if any, of #f(x)=3x^5 - 5x^4 #?

Answer 1

Aurora Alvarado

The only inflection point is #(1,-2)#.

A point on the graph where the inflection (concavoty) changes is called an inflection point.

#f(x) = 3x^5-5x^4#

#f'(x) = 15x^4-20x^3#

#f''(x) = 60x^3-60x^2#

# = 60x^2(x-1)#

Ths sign of #f''# changes at #x=1#, so the point #(1,f(1))# is an inflection point.

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Answer 2

Evelyn Agard

To find the points of inflection, we need to find the second derivative of the function and set it equal to zero.

( f(x) = 3x^5 - 5x^4 )

First derivative:

( f'(x) = 15x^4 - 20x^3 )

Second derivative:

( f''(x) = 60x^3 - 60x^2 )

Setting the second derivative equal to zero:

( 60x^3 - 60x^2 = 0 )

( 60x^2(x - 1) = 0 )

( x = 0 ) or ( x = 1 )

So, the points of inflection are ( x = 0 ) and ( x = 1 ).

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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.