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

##### 1 Answer

when

#### 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)=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

#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

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

When

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

Since the sign of the second derivative *does* change around

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

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