What are the points of inflection, if any, of #f(x)= 9x^3- 5 x^2-2 #?

1 Answer
Jan 13, 2016

when #x=5/27#

Explanation:

A point of inflection occurs when the second derivative of a function switches sign (goes from positive to negative, or vice versa).

Find #f''(x)#:

#f(x)=9x^3-5x^2-2#
#f'(x)=27x^2-10x#
#f''(x)=54x-10#

The second derivative could from positive to negative or negative to positive when #f''(x)=0#. Find those points:

#54x-10=0#

#x=10/54=5/27#

Check to make sure the second derivative actually changes sign around this point. As of now, it is just a possible point of inflection.

When #x<5/27#, we can test #f''(0)#:

#f''(0)=-10" ... "<0#

When #x>5/27#, we can test #f''(1)#:

#f''(1)=44" ... ">0#

Since the sign of the second derivative does change around #x=5/27#, it is a point of inflection.

We can check this graphically--the concavity should shift.

graph{(9x^3-5x^2-2) [-10, 10, -7.2, 2.8]}