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

To determine the point of inflection for the given function y = x^3 - x^2 - 8x + 12, we need to find the second derivative.

Step 1: Find the first derivative of the function.
To do this, take the derivative of each term:
dy/dx = d/dx(x^3) - d/dx(x^2) - d/dx(8x) + d/dx(12)

The derivative of x^n (where n is any constant) is nx^(n-1).

dy/dx = 3x^2 - 2x - 8

Step 2: Find the second derivative.
Now, take the derivative of the previous result:
d^2y/dx^2 = d/dx(3x^2 - 2x - 8)

Differentiating each term:
d^2y/dx^2 = d/dx(3x^2) - d/dx(2x) - d/dx(8)

The derivative of constant K is 0, and the derivative of bx (where b is any constant) is b.

d^2y/dx^2 = 6x - 2

Step 3: Set the second derivative equal to zero and solve for x.
To find the potential points of inflection, we need to find where the second derivative is equal to zero.
Set d^2y/dx^2 = 0:

6x - 2 = 0

Solving for x:
6x = 2
x = 2/6
x = 1/3

Step 4: Test the interval around x to determine the concavity.
We need to check the concavity on both sides of x = 1/3 to determine if it is an inflection point. To do that, we can take any value less than 1/3 and any value greater than 1/3 and substitute them into the second derivative expression.

For example, let's choose x = 0 (less than 1/3) and x = 1 (greater than 1/3).

When x = 0:
d^2y/dx^2 = 6x - 2 = 6(0) - 2 = -2
Since the second derivative is negative, the graph is concave down.

When x = 1:
d^2y/dx^2 = 6x - 2 = 6(1) - 2 = 4
Here, the second derivative is positive, indicating the graph is concave up.

Based on this analysis, since the concavity changes at x = 1/3, it is the point of inflection for the given function y = x^3 - x^2 - 8x + 12.

Thus, the point of inflection is (1/3, f(1/3)), where f(1/3) is the value of the function at x = 1/3.