Question

Find all the local and absolute maximums and minimums

Answer 1

http://roy.math.umn.edu/webwork2_course_files/umtymp-calculus1-f13/tmp/gif/muzzammil-Q-raza-2211-setExtrema_and_MVTprob1image1.png

Answer 2

Ok, so where do you think these suckers occur? Local means that nearby they're the lowest or highest points. Absolute means they're the highest or lowest of all the points.

Answer 3

i think that the local max is at x=3, local min at x=2 and x=5, i can't find the absolute max, the absolute min at x=5

Answer 4

@Kainui

Answer 5

That seems about right, but what about at x=8?

Answer 6

x=8 isn't a local or absolute max because local maxs can't be endpoints and its not absolute because there are bigger numbers. what would the absolute max be for this function?

Answer 7

There is no absolute max, is there? If you pick the endpoint at x=0, it's not included, and if you pick any point near there I can always pick [6+(that number you pick)]/2.

Answer 8

this is online hw so it tells me if it is right or wrong. it says something is wrong with these answers
local max: 3
Local min: 2, 5
abs max: None
abs min: 5

Answer 9

Well I guess I'm wrong, but that seems right to me.

Answer 10

Perhaps your local min at 2 is incorrect, not sure if counts as being "local".

Answer 11

muzzammil: I'll respond to your earlier statements first:
a) I agree you have a local max at (3, 5) (that is, at x = 3, y = 5).
b) Local minimum at x = 2? No, because the function (and thus the graph) is discontinuous at x = 2.
c) At x = 5, the function is continuous and takes on the smallest possible y value. Thus, you have an ABSOLUTE minimum at (5, 1). This is not a local minimum, however, for at least two reasons: first, the function is not differentiable at x = 5, and second, the slope is not zero there (zero slope is manadatory at a local min. or max.)
d) (0,6) is not an absolute max because the function is not defined at x = 0 (see that open circle?)
e) We do have an absolute max at x = 3, since y is defined there. In other words, (3,5) represents both a local and an absolute max.

Answer 12

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Answer 13

ok so @mathmale what would the local and absolute min be

Answer 14

muzzammil: Look for the smallest defined y value on the given interval. That point represents the absolute minimum of the function on this interval.
Now, is there a local minimum? Do you see any interior point on the graph (i. e., not an endpoint) where the slope is zero? No? What can you conclude about the presence or absence of a local minimum?

Answer 15

there is no local min but the absolute min is at x=5?

Answer 16

(Applause)

Answer 17

where is the absolute max?

Answer 18

Look for the (interior) point on the graph where the function achieves a max AND the slope of the tangent line to the graph is zero. That represents the local maximum.

Answer 19

What about the Absolute max

Answer 20

To be honest, I have some qualms about that one. On the left, at (0,6), the function is NOT DEFINED, right? Were it defined, then that point would definitely be the ABS MAX. But undefined, that point cannot be an ABS MAX. So, what would your conclusion be?

Answer 21

In other words, aside from (0,6), is there any other point on the graph at which the function is defined and has the largest defined value on this entire interval [0,8]?

Answer 22

my conclusion is i don't know

Answer 23

As I said before, I was least certain of my decision that (0,6) is NOT an absolute max, for the reasons already given. On the other hand, (3,5) does represent an abs. max. (as well as a local max.) because it is the largest y value on the interval (0,8] at which the function is defined. Thus, I'd conclude: abs. and loc. max. at (3,5).

Answer 24

and there is no local min?

Answer 25

That'd be my conclusion also. Can you explain why there's no local min.?

Answer 26

no because there should be a local min

Answer 27

On what would you base your statement that there should be a local minimum? The definition of "local minimum" is pretty fussy: the point must be within the interval (a,b); the function must be differentiable there; and the slope of the tangent line there must be zero.

Answer 28

Are there any such points on your graph? I think not.