Critical Numbers / Shelf / Cusp

[Math007]

Hello,

I have a table of values and am asked to find between what values does the function have critical numbers. I will give a small portion of the table to use as an example:

x | - 3 - 2 - 1 0 1 2
f(x)| - 20 - 4 5 1 -1 5

Now if you graph this function, you will see that there is a local max at -1 and a local min at 1.

My question is whether I got the correct answer and if there is another alternative to doing it rather than graphing it. Also would -3 and 2 be critical numbers because they are ther absolute max and min?

Finally, I have a question about shelfs and cusps. When do they occur on a graph? I think that shelfs occur when the sign of f'(x) goes + 0 + or - 0 - or alternatively the graph of f'(x) goes towards the x axis and touches it and then comes back down without crossing it. Also, I think that cusps occur at values of f'(x) that are undefined. But i am really not sure.

Thanks for the help. I really appreciate this great service you guys offer!
Math007

 

[pka]

“Now if you graph this function, you will see that there is a local max at -1 and a local min at 1.”
First, we do not know that a local max is at exactly –1. However we do know that between –1 and 0 there is a critical point. The same is true for 1.
The way to do this is to note the intervals of increase and decrease.

 

[Math007]

Thanks for correcting that error. I guess it's impossible to know where it exactly occurs unless you know the function and differentiate it and set it to zero. Also would the two endpoints that are both a global max and min be considered critical points?

Thanks
Math007

 

[pka]

would the two endpoints that are both a global max and min be considered critical point?

There is no way to know about global properties from the data.

 

[Math007]

Hello,

Thanks for the help. Could you also help clarify something for me. If say you have a parabola, would the vertex be called the absolute min/max (whchever it is) or would it be a local max. Would it be incorrect to call the vertext a local max/min?

Thanks

 

[skeeter]

for a parabola, the vertex will be the location of an absolute extremum.

all absolute extrema are local extrema, but not all local extrema are absolute.

 

[Math007]

I have an equation: x^4/4 - 7x^3/3+8x^2-12x

I found the derivative which is (x-2)^2(x-3) and I did a sign chart:

x <=2 = negative
2<x<3 = negative
x > 3 = positive

i know that x = 3 is a local minimum (and absolute) because the sign changes from negative to positive. But I am unable to identify what x = 2 is since the sign goes negative negative and there is no change.

Help would be appreciated.
Thanks!