Use the second derivative to determine the point of inflection for y = x3 – x2 – 8x + 12. ????

To determine the point of inflection for the function y = x^3 – x^2 – 8x + 12 using the second derivative, you need to follow these steps:

Step 1: Find the first derivative:

To find the first derivative, you differentiate the function y = x^3 – x^2 – 8x + 12 with respect to x.
dy/dx = 3x^2 – 2x – 8

Step 2: Find the second derivative:

To find the second derivative, differentiate the first derivative with respect to x.
d^2y/dx^2 = d/dx (3x^2 – 2x – 8)
d^2y/dx^2 = 6x – 2

Step 3: Set the second derivative to zero and solve for x:

To find the point of inflection, set the second derivative equal to zero and solve for x.
6x – 2 = 0
6x = 2
x = 2/6
x = 1/3

Step 4: Plug the value of x into the original function to find the y-coordinate:

Plug the value of x = 1/3 into the original function y = x^3 – x^2 – 8x + 12 to find the y-coordinate.
y = (1/3)^3 – (1/3)^2 – 8(1/3) + 12
y = 1/27 – 1/9 – 8/3 + 12
y = 1/27 – 3/27 – 72/27 + 324/27
y = 250/27

Therefore, the point of inflection for the given function is (1/3, 250/27).