Alaina's blog, 14 November 2010

This week, we talked about the second derivative test. It has most of the same steps as the first derivative test, and then some of its own.
First, you take the first derivative, set it equal to zero, and solve for x.
Then, you take the second derivative, set it equal to zero, and solve for x.
Next, you check for differentiability. If it is not differentiable, the second derivative test fails.
Then, you set up intervals with your x value from the second derivative.
Next, you plug values from your intervals into the second derivative.
If the output value is positive, it is concave up. If the output value is negative, it is concave down.

Ex: Determine the intervals on which the graph is concave up or concave down for f(x)=(-x^3)+(6x^2)-9x-1
**because it is only asking for the intervals on which the graph is concave up or down, you do not need to complete step 1 (solve the first derivative for x).
1. f’(x)=(-3x^2)+(12x)-9
2. f”(x)=(-6x)+12 = 0
-6x=-12
x=2
3. yes it is differentiable
4. (-infinity, 2)u(2, infinity)
5. f”(0)=(-6(0)^2)+12= positive, concave up.
f”(3)=(-6(3)^2)+12=negative, concave down.
The function is concave up on the interval (-infinity, 2) and concave down on the interval (2, infinity).