Question

What are the points of inflection of #f(x)=x^3 - 9x^2 + 8x #?

Answer 1

at `x=3`

Explanation:

Points of inflection occur when the second derivative of a function shifts sign (goes from positive to negative). This can also be viewed as a shift in concavity.

Find the second derivative:

`f(x)=x^3-9x^2+8x`
`f'(x)=3x^2-18x+8`
`f''(x)=6x-18`

The second derivative's sign could shift when it equals zero.

`f''(x)=0`
`6x-18=0`
`x=3`

There is a possible point of inflection when `x=3`.

We can check by analyzing the sign around the point `x=3`.

`(-oo,3)`

`f''(0)=-18larr"negative"`

`3`

`f''(3)=0larr"zero"`

`(3,+oo)`

`f''(4)=6larr"positive"`

As you can see, the concavity shifts from negative (concave down) to positive (concave up) when `x=3`. Thus, it is a point of inflection.

`f(x)` graphed:

graph{x^3-9x^2+8x [-3, 10, -78.3, 27.1]}