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]}