Is the point of inflation of the function f(x)= 2x^3 - 12x^2 + 8x -4 at (2, -20)

I just want to make sure I got it right.

Inflection points occur when the second derivative is 0, so first we find the second derivative:

f(x) = 2x^3 - 12x^2 + 8x - 4

By the power rule:

f '(x) = 6x^2 - 24x + 8

Again by the power rule:

f ''(x) = 12x - 24

So, we have our second derivative, and we let it equal 0 to find the x-coordinate of the (in this case only) inflection point:

12x - 24 = 0

Add 24:

12x = 24

Divide by 12:

x = 2

So, our x-coordinate is x = 2, plug this into the original equation to find the y-coordinate at this point:

f(2) = 2(2)^3 - 12(2)^2 + 8(2) - 4

Solve:

f(2) = 2(8) - 12(4) + 16 - 4

f(2) = 16 - 48 + 12

f(2) = -20

So, the point is (2, -20), you are correct!