4.4 Concavity and Inflection Points

Wed Oct 21Do Now

Find the 2nd derivative of each function

1)

2)

Applications of the 2nd derivative

• So far, we’ve only talked about one application of the 2nd derivative, which is the acceleration function

• The second derivative can also be used to describe the behavior of functions as well.

Concavity and Inflections

• The 1st derivative is used to describe slope. But since it is also a function, it also has its own “slope” or derivative.

• The 2nd derivative can be used to model the behavior of the slope, as it is ALSO changing with the function– Some slopes can be steep, while others

rather flat

Concavity• The 2nd derivative can be used to

describe concavity

• Concavity is the rate at which the slope increases or decreases

• There are two types of concavity– Concave up (looks like a smile)– Concave down (looks like a frown)

Concavity and f’’(x)

• Thm- Suppose f(x) is differentiable on an interval I and f ’’(x) exists, – If f ’’(x) > 0, then the graph is concave up– If f ’’(x) < 0, then the graph is concave

down

Note: Second derivative only

Inflection Points

• An inflection point is a point on the graph where a graph alternates between concave up and concave down

• We can find inflection points when f ’’(x) = 0

Example 1

• Determine where the graph is concave up and concave down

Example 2

• Determine where the graph is concave up and down, and find any inflection points

Ex 5.3

• Determine the concavity and inflection points of

2nd Derivative Test• The 2nd Derivative can also be used to

determine if a critical point is a local max or min.

• Thm- Suppose that f(x) is continuous on an interval (a,b) and f’( c) = 0, then– If f’’( c) < 0, then c is a local max

• Concave down means a local max!

– If f’’( c) > 0, then c is a local min • Concave up means a local min!

Warning!

• The 2nd derivative test does not always work.

• It will not work if f’’(c) = 0

• If the 2nd derivative test does not work, you must use the table

Ex

• Analyze the critical points of

Ex 2

• Use the 2nd derivative test to find the local max and mins for

Closure:

• Hand in: Find intervals of increase, decrease, and concavity, local extrema, and inflection points of

• HW: p.239 #1 5 13 25 33 37 43 47 53-56 65

• 4.1-4.4 Quiz tomorrow