Question

Find all maximum and minimum points of y=x+sinx on the interval -pi/2<=x<=3pi/2.
I have found that y' is positive for all numbers but is 0 at pi. So is that correct and how would I find the max/min?

Answer 1

So, you got to the point where y'=1+cosx, right?

Answer 2

yes

Answer 3

Now, at the max/min point, the slopes are 0.

Answer 4

So, first, you set y'=0 to solve over the interval of [pi/2,3pi/2].

Answer 5

so 1+cosx=0
x=0?

Answer 6

cosx=-1
x=pi

Answer 7

ohh right. now what?

Answer 8

We know that at x=pi, it's either maximum or minimum.
Do you know how to determine whether it's maximum or minimum?

Answer 9

of f' changes sign from positive to negative its a max, negative to positive is a min.
but i get positive for everything. i plugged in -pi/2 and 3pi/2 into 1+cosx

Answer 10

Remember that the interval is from pi/2 to 3pi/2, inclusive.

Answer 11

the interval is -pi/2 to 3pi/2. sorry, in the question the negative sign didn't stay with the pi

Answer 12

Oh.

Answer 13

But, it doesn't really change the question.

Answer 14

Now, the reason you got all positive is simple.
Graph y=cosx on your scratch paper from -pi/2 to 3pi/2

Answer 15

ok

Answer 16

Notice how from -pi/2 to pi/2, it's completely positive while from pi/2 to 3pi/2, it's completely negative.

Answer 17

Here, we learn a lesson; to pick close points.
So, 3pi/4 and 5pi/4 could be nice points that are close.

Answer 18

But, there is an easier way without substituting point.
Here is how.

Answer 19

Have you ever heard of second derivative?

Answer 20

yes

Answer 21

Can you tell me what it is?

Answer 22

after you get y', you take the derivative of that to get y''

Answer 23

Good. Now, do you know the meaning of the second order derivative?
We know that the first order derivative is the slope.

Answer 24

nope, i've never heard of second order derivative

Answer 25

Ok. Second order derivative describes the slope of the slope.
In other words, the rate of change for slope.

Answer 26

hmm ok

Answer 27

It'll get clarified with this example.
What is the second order derivative of y=x+sinx?

Answer 28

y' = 1+cosx or are you asking for 2nd derivative? which would be y''=-sinx

Answer 29

2nd one, which is correct.

Answer 30

alright.

Answer 31

Now, if the second derivative is negative, then it would mean that the slope decreases, right?

Answer 32

If slope decreases, the graph of y=x+sinx would increase, right?

Answer 33

It looks like that if the slope decreases.

Answer 34

hmm i'm not really understanding. all i know is that in the picture, it's concave down, meaning the second derivative is negative

Answer 35

Exactly!

Answer 36

Sorry for replying a bit late.

Answer 37

If the graph looks like on the left, then can you imagine what it would look on the right, assuming there is an extremum.

Answer 38

sorry, what do you mean?

Answer 39

Can you draw what it would look like on the right?

Answer 40

nope im not getting it

Answer 41

Look at your cosine graph from -pi/2 to pi/2.
Look familiar?

Answer 42

yeah.. so the right side would be going down, concave up?

Answer 43

Correct!

Answer 44

Now, what happens the second order derivative is positive?

Answer 45

the function is concave up and the first derivative is increasing

Answer 46

So what does the graph look like?

Answer 47

the graph of y' or y'' ?

Answer 48

or y ?

Answer 49

y

Answer 50

y is increasing up to pi , where it equals 0, and continues increasing? and its concave up

Answer 51

cos(pi)=-1

Answer 52

ohh...i keep making that mistake! :(

Answer 53

After understanding how to use second order derivative to find extrema, there is something I have to tell you that I just found out.
So, first, what does y look like when second order derivative is positive?

Answer 54

y is concave up

Answer 55

Actually, y is concave down.