Question

I had a question, i was reading in my calc book the chp on curve tracing, and it made a statement i quite don't comprehend fully. It said that when y' is 0 we do not know whether y is increasing or decreasing. And then it said that when y' is 0 usually y has a rel. max/min. I am confused because i thought that when a graph as a rel. max/min we could determine whether that function is increasing or decreasing when or at the points where the derivative is 0. could someone clarify?

Answer 1

usually to determine concavity look at y''.
If y'' >0. y' is increasing, rel min
y'' <0 . y' is decreasing, rel max

Answer 2

but when y' is zero, we can dtermine if a function is increasing or decreasing correct?

Answer 3

true, thats why we look at y''
for x=1, say y'=0
well at x=1, y'' = -2
this tells us we have a rel max, function is going from increasing to decreasing

Answer 4

okay

Answer 5

you can also evaluate at points less than critical point to determine increasing/decreasing
so at x=1, y'=0, y=5
look at x=0, say y' = 3, y=3
then y is increasing, y' is decreasing
thus rel max

Answer 6

critical points would be where the derivative is 0 right?

Answer 7

yes

Answer 8

it depends on the function, if it it easily differentiable, take 2nd derivative otherwise use other approach

Answer 9

let me asking you something else, regarding abso and rel max and min. Lets say i have a cubic function and the derivative was 0 at x=1 and x=3. And i determined that that function has a rel max of 3(1/3) at x=1. But now i want to use an interval of [0,3], why does my book say that the abso. max is still 3(1/3), and that there are abso. min values at two locations 0 and 3.

Answer 10

because you have restricted the domain,
the section of graph from [0,3] has abs min at 0,3 and abs max at 1
however that is not true for the function just the part of the graph from [0,3]

Answer 11

its like you defined a new function, where 0<x<3

Answer 12

When it comes to working with a closed interval, my book says that the absolute mx and min must occur at either on or both ends of the intervlal or it must occur at some value of x within the interval. Now, does that mean, that if i have determed that there is a rel. max or min, at an interrior point, lets say that [0,3], x=2, then that y value at x=2 would be the abso. max or min as well as the rel max or min?

Answer 13

yes, many times the rel max/min is the abs max/min in a specified interval but it depends on the interval

Answer 14

for example if the interval is wide enough it may be the end point has larger y value than the rel max, thus the abs max would be the end point

Answer 15

So i should evlaute both the endpoint of the interval. Now what happens if the derivatve of a function is 0 at point within the interval and i determined that at either one of these points there is a max or min. Noe would i still need to check the endpoints?

Answer 16

oh okay, so if i found that the derivative is zero at interior points and i determined that they give me a max or min, but then i also evaluate the endpoint, and find that the endpoint give me higher (max) or lower(min) values, these would then be the abso max/min

Answer 17

yes

Answer 18

I appreciate your patience, i am taking calc over the summer so that i can finish up my 2 years at community college quicker and that why i am asking these questions. Thanks again.

Answer 19

no problem, good luck with your class