1. The 2nd Derivative

2. The derivative of the first derivative
What is it???
The derivative of the first derivative

3. What is the notation?

4. Example:

5. What can we learn from the 2nd Derivative
We can find out if a stationary point is a local maximum or a local minimum

6. Reminder: What is a stationary point?
A point (a, f(a)) where f’(a)=0
In other words a point on the function f(x) where the derivative is equal to zero

7. SECOND DERIVATIVE TEST
How do we use the 2nd Derivative to determine whether a stationary point is a local minimum or a local maximum?
SECOND DERIVATIVE TEST

8. Second Derivative Test
The test says that if a function f(x) is differentiable at a stationary point x=a, meaning that f’(a)=0, then:

9. Second Derivative Test
The test says that if a function f(x) is differentiable at a stationary point x=a, meaning that f’(a)=0, then:
If f’’(a)<0 then f(x) has a local maximum at x=a

10. Second Derivative Test
The test says that if a function f(x) is differentiable at a stationary point x=a, meaning that f’(a)=0, then:
If f’’(a)<0 then f(x) has a local maximum at x=a
If f’’(a)>0 then f(x) has a local minimum at x=a

11. Second Derivative Test
The test says that if a function f(x) is differentiable at a stationary point x=a, meaning that f’(a)=0, then:
If f’’(a)<0 then f(x) has a local maximum at x=a
If f’’(a)>0 then f(x) has a local minimum at x=a
If f’’(a)=0 then the second derivative doesn’t tell us what type of stationary point x=a is and we need to look at the sign diagram

13. What else can the 2nd Derivative tell us?
In relation to rate of change