Question

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

Answer 1

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]}