# What are the points of inflection, if any, of #f(x)=3x^4 − 6x^3 + 4 #?

##### 1 Answer

They occur at

#### Explanation:

Points of inflection are where a function's concavity shift from concave upwards to concave downwards, or vice versa.

Since concave upwards functions occur when the second derivative is positive, and concave downwards correlates to the second derivative being negative, a point of inflection occurs at the switching point when the second derivative equals zero.

So, we need to set the second derivative equal to

The second derivative can be found through applying the power rule:

#f(x)=3x^4-6x^3+4#

#f'(x)=12x^3-18x^2#

#f''(x)=36x^2-36x#

Points of inflection could occur when:

#36x^2-36x=0#

#36x(x-1)=0#

So, when

Test the sign of the second derivative around these points:

#f''(-1)=36(-1)(-1-1)=72>0#

#f''(0)=0#

#f''(1/2)=36(1/2)(1/2-1)=-9<0#

#f''(1)=0#

#f''(2)=36(2)(2-1)=72>0#

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