Lindsey's Reflection

This week we left chapter 5 and went
back to chapter 3 again.

we learned about the second derivative test
which is basically the same thing as the first derivative test
except that there are a few changes in the steps.

here are the steps for the second derivative
test:

1. instead of taking just the first derivative
you take the second derivative and then you
set it equal to zero and solve for x.

2. find anywhere that it is not differentiable.

3. step up intervals with the numbers
that you found in step one and two.

4. plug in a number from each of the
intervals into the second derivative
equation.

5. if it is positive it is concave up
and if it is negative then it is
concave down.

6. if it changes from + to - or
- to + there is a point of inflection.
so if it is - - or ++ there are no
points of inflection.

so those are the few differences.

here is an example using this concept:

f(x)= 6/x^2+3

determine the open intervals where
the function is concave up or concave down.

1. f^1(x)= -12/(x^2=3)^2

f^11(x)= 36(x^2-1)/(x^2+3)^2

36(x^2+3)=0

x= +/- 1

2. x^2+3=0

x= +/- Square root of -3

3. (-infinity, -1) u (-1,1) u (1,infinity)

4. f(-2)= + f(0)= - f(2)= +

5. x= -1 is a point of inflection
x= 1 is a point of inflection

concave up: (-infinity,-1) u (1,infinity)

concave down: (-1,1)