In this section we get word close to the ii types of curvature and determine the curve of a function.

Concavity

In this section we will discuss the curvature of the graph of a given function. There are two types of curvature: saclike up and cuplike down. The main tool for discussing curvature is the second derivative, .

The following theorem helps us to determine where a function is concave up and where it is cup-shaped down.

The last example brings up a new concept. The operate changes concavity at . We call this an inflection place of the function.

In the next ii examples, we will discuss the curvature of the given functions and bump their inflection points.

(problem 4a) Find the inflection point(s) of the routine

Find the second derivative,

Where does change sign?

Is day-and-night there?

The function has an inflection point at

(job 4b) Find out the inflection point(s) of the function

Find the second first derivative,

Where does interchange sign?

Is continuous there?

The function has an inflection repoint at

(problem 5a) Find the inflection point(s) of the mathematical function

Find the s derivative,

Where does change sign?

Is continuous there?

List multiple answers in ascending order.
The function has inflection points at and .

(problem 5b) Find the inflection point(s) of the serve

Find the second derivative,

Where does change sign?

Is continuous thither?

If thither are no inflection points, typecast "none".
The function has flection point(s) at

(problem 5c) Feel the intervals of increase/decrease, local extremes, intervals of concavity and inflection points for the routine

(trouble 6a) Observe the inflection taper(s) of the subprogram

Find the second derivative,

Use the product rule to work out the derivatives

and is similar

Where does change sign?

Is continuous at that place?

The function has an inflection point at

(problem 6b) Find the inflection period(s) of the social function

Find the 2d derivative,

Use the product rule to compute the derivatives

and is kindred

Where does change sign?

Is continuous there?

List multiple answers in ascending order.
The function has inflection points at and

(trouble 6c) Notic the inflection show(s) of the function in the interval

Find the indorse differential,

Where does convert sign on?

Is continuous there?

Tilt fourfold answers in ascending order.
Happening the interval , the function has inflection points at and .

(problem 7) Find the flection point(s) of the part

Find the second derivative,

Use the Quotient Dominate to work out the derivatives

Where does ?

To solve the equation ,
let and solve for kickoff

Is continuous on that point?

List multiple answers in ascending order.
The function has inflection points at and .

(problem 8) Find the inflection point(s) of the function on the interval .

Find the second derivative,

The differential of is

Use the Chain Rule to breakthrough the derivative of

Where does ?

Along the interval , the function
has an inflection steer at .

Using the graph of the derivative

In this section, we are given the graphical record of the derivative, , and we are asked to cause conclusions about the original function, .

(trouble 9) Use the graph of the derivative, , given below, to answer the favorable questions.

Where is increasing? Select all that apply.

Where is decreasing? Select completely that hold.

Describe the local extremes of . Select all that practice.

local level bes at local anaesthetic upper limit at topical anesthetic minimum at local minimum at local anesthetic marginal at

Where is concave up? Select all that apply.

Where is concave down? Prime completely that practice.

Where are the prosody points of ? Choice every that use.

inflection channelis at inflection point at inflection point at nobelium flexion points

(problem 10) Use the graph of the derivative, , given below, to solvent the following questions.

Where is increasing? Select all that apply.

Where is depreciative? Select whol that apply.

Describe the local anesthetic extremes of . Select all that apply.

local maximum at local maximum at local minimum at local lower limit at local minimum at

Where is concave astir? Select all that lend oneself.

Where is concave down? Take all that go for.

Where are the inflection points of ? Select all that apply.

inflection channelize at inflection point at modulation point at no prosody points

Video lessons

Hera is a detailed, trounce title video on concave shape:

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We following explore a special relationship 'tween concavity and local extremes.

The following figure should convince the reader of the rigor of the Second Derivative Test.

(problem 11) Use of goods and services the Bit derivative Test to find the localized extremes of The critical numbers are (list in ascending order) and .
has a localized maximum at and a local anaesthetic minimum at