Determine the points of inflection of y=x^3-3x^2-9x+6.

The inflection points could be found by calculating the
roots of the second derivative of the function (if there are
any).

For the beginning, we'll differentiate the
function:

dy/dx=d/dx(x^3-3x^2-9x+6)

dy/dx
= d/dx(x^3) - d/dx(3x^2) - d/dx(9x) + d/dx(6)

dy/dx = 3x^2
- 6x - 9

Now, we'll differentiate
dy/dx:

d^2y/dx = d/dx(3x^2 - 6x -
9)

d^2y/dx = d/dx(3x^2) - d/dx(6x) -
d/dx(9)

d^2y/dx = 6x -
6

or

f"(x) = 6x -
6

After f"(x) calculus, we'll try to determine the roots of
f"(x).

f"(x) = 0

6x - 6 =
0

We'll divide by 6:

x - 1 =
0

x = 1

For x = 1, the
function has an inflection point.

f(1) =
1^3-3*1^2-9*1+6

f(1) = 1 - 3 - 9 +
6

f(1) = -5

The
inflection point is: (1 , -5).