Question

The figure below shows the graph of f ′, the derivative of the function f, on the closed interval from x = −2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4.

Find the x-value where f attains its absolute minimum value on the closed interval from x = −2 to x = 6. Justify your answer.

Answer 1

https://gyazo.com/2c47b68b177653bc9680c8043feceba6

Answer 2

@terenzreignz @jim_thompson5910 @freckles Absolute minimum is lowest point on the graph?

Answer 3

at 5 the derivative goes from being negative (below the x axis) to positive (above the x axis)
that means the function itself goes from decreasing to increasing

Answer 4

`Absolute minimum is lowest point on the graph?`

correct @Ephemera

Answer 5

Ok so I would need to find the absolute minimum of f using the graph of f'?

Answer 6

you're in a calculus class, so naturally you should use calculus

Answer 7

find the local extrema and also test the endpoints

Answer 8

Say what?

Answer 9

Wasn't taught that so you're gonna have to allow me a few minutes to google it.

Answer 10

hold on, I remember this problem a while back

let me find it

Answer 11

Unless it's not needed.

Answer 12

haha
you probably were, just that different wording was used :D

Maximums or Minimums, they are collectively called Extremums

Let's call a few points "extremum candidates"

these are either endpoints of the interval of places where the DERIVATIVE is zero.

I see four such candidates, can you identify them?

Answer 13

@jim_thompson5910 you helped me with a similar one a while back and I was still a little confused that's why I reposted.

Answer 14

take a look at my picture, it should be clear

Answer 15

derivative \(f'\) is negative, function\(f\) is decreasing (going dow)
derivative is positive, function is increasing (going up)
where it changes from deceasing to increasing you have a local minimum

Answer 16

(-2,-3)
(2,0)
(5,0)
(6,4)

Answer 17

see this page

http://openstudy.com/study#/updates/56638a0fe4b03c442cf28a29

Answer 18

@satellite73 I see

Answer 19

Just the x-values will do.

Answer 20

Now, look at the values x = 2 and x = 5
Check how the derivative behaves around the said points.

before x = 2, is the derivative positive or negative?
what about after?

Answer 21

Ok got it. So I look at a point where the derivative is 0 and look at its surroundings to determine if it's a minimum or not.

Answer 22

So for a min before it would be negative and after positive while the max would be the opposite?

Answer 23

Yes. For it to be a minimum, it had better be decreasing BEFORE the x-value in question and increasing AFTER.

Answer 24

Yes, exactly :D

Answer 25

Gr8 8/8 m8 much appreci8

Answer 26

Not quite yet. We have only ever shown that a relative minimum exists on that x=5 point.
Not yet an absolute minimum.

Answer 27

So... we can rule out x=2 as the function was decreasing both before and after x=2
What about the end points?
The right end-point, x=6.
Could the function possibly be lower at x=6 than it is at x=5?

Answer 28

No, As it is above the x-axis by a mile compared to x=5

Answer 29

"No" is correct but the justification is not quite there ^^
The reason is that the derivative has been positive for the entire duration on the interval (5,6]. That means the function has only gone up since then.

What about the other endpoint, x=-1 ?
Could the function be lower at x=-1 than it is at x=5?

Answer 30

Do you mean x=-2?

Answer 31

Sorry. Yes, I do mean x=-2 :)

Answer 32

Well it can't be because it was always negative at that point. ;)

Answer 33

Or the area surrounding that point is also negative.

Answer 34

haha
the derivative has always been non-positive since it went from -2 to 5.

that means the function has never increased since it was from -2 up until it got to 5.

that means that, yes, the function is higher on -2 than it is on 5.

So no, -2 is not the absolute minimum

telling us conclusively that x=5 is at the ABSOLUTE MINIMUM of the function on the interval [-2 , 6]


FINALLY DONE lol

Answer 35

Now 8/8?

Answer 36

8/8 m8 gr8 appreci8

Answer 37

No problem :D