Relating the Graphs of f, f’ and f’’
First Derivative Test for
Local Extrema

 TBT: What is a critical point of a function?
 Point at which f’ is zero or undefined.
 At a critical point c…..
 1. If f’ changes sign from positive to negative at c,
then f has a local maximum value at c.
First Derivative Test

 2. If f’ changes sign
from negative to
positive at c, then f has
a local minimum at c.
 3. If f’ does not change
sign at c (same sign on
both sides), then f has
no local extreme value
at c.
How can you tell if an endpoint
is a max or min numerically?

 Analyze the slope near the endpoints.
Examples

 (a) Find local max and mins.
 (b) Identify the intervals in which the function is
increasing or decreasing.
 1. f(x) = x3 – 6x2 + 9x + 1
 2. 𝑓 𝑥 =
ln 𝑥
𝑥3
The Second Derivative

 What does y’’ tell you about the graph of y’?
 When y’’ is positive, y’ is increasing.
 When y’’ is negative, y’ is decreasing.
 What does y’’ tell you about the graph of y?
 When y’’ is positive, the slope of y is increasing.
 When y’’ is negative, the slope of y is decreasing.
 What does increasing/decreasing slope look like?
 When y’’ is positive, y is concave up.
 When y’’ is negative, y is concave down.
Concavity on a Graph

 Where does the concavity change on the graph
below? These points are called points of inflection
and they are critical points for y’’.
Examples

 3. Use the function from example 1:
 f(x) = x3 – 6x2 + 9x + 1
 Determine where this function is concave up and
concave down.
 Combine the information learned from f’’ with the
info learned from f’ into one chart.
 Sketch the graph of f(x) using the information in the
chart.
Additional Example

 f(x) = x2ex
 1. Find the x-coordinates for any critical points.
 2. Determine where f is increasing/decreasing.
 3. Determine the max and mins.
 4. Find where the function is concave up/down.
 5. Locate any points of inflection.
 6. Make a rough sketch based on your answers from
1-5.
Application to Motion

Position = s(t)
Velocity = v(t) = s’(t)
Acceleration = a(t) = v’(t) = s’’(t)
Remember that these are all vectors. Positive values
generally indicate to the right or up. Negative values
usually refer to left or down.
 Speed is not a vector. If acceleration and velocity have the
same sign, the object is speeding up (either in a + or –
direction.
 If a(t) and v(t) have opposite signs, the object is slowing
down.




Example

s(t) = 3t4 – 16t3 + 24t2
1. When is the object moving left/right?
2. When does the object reverse direction?
3. When is the velocity increasing/decreasing?
4. When is the object speeding up or slowing down?
5. Describe the motion of the object (including its speed)
in words
 6. Sketch a graph of the position curve.
 7. Sketch a graph of the velocity curve.
 8. Verify answers using the graphing calculator.

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Examining Critical
Values

𝑥
𝑦
 Suppose y’ = . What can you tell me about the
point (0, 4)?
 Why can’t you determine if it is a max or a min?
 We don’t have other values for y’ at nearby points.
 Second Derivative Test for Local Extrema
 If f’(c) = 0 and f’’(c) < 0, then f has a local maximum
at x = c.
 If f’(c) = 0 and f’’(c) > 0, then f has a local minimum
at x = c.