Home > Calculus > Determining Points of Inflection for a Function

How do you find the inflection points of #f(x)=3x^5-5x^4-40x^3+120x^2#?

Answer 1

The inflection points are #x=-2,1,2#

To find the inflection points you need to perform the second derivative test. Since this is a polynomial we use the power rule to differentiate the equation, #nx^(n-1)#. We get the first #d/dx# which is:

#f'=15x^4-20x^3-120x^2+240x#

Followed by the second #d/dx# which is:

#f''=60x^3-60x^2-240x+240#

Now we factor you should get:

#60x^2(x-1)-240(x-1)#

#(60x^2-240)(x-1)#

Now set the factors equal to zero:

#60x^2-240=0# and #x-1=0#

Solve them and you should get:

#x=+-2, 1#

If you wish to find out where the exact inflection points occur plug in the three values into the original equation :) You would do #f(-2), f(1), f(2)#.

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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.