Calculus
posted by Janet on .
Find any absolute maximum and minimum and local maximum and minimum for the function f(x)=x^34x+5 on the interval [0,5). Make sure to prove that these points are max/min values.
THis is what I did.
f'(x)=3x^24=0
x= +/ 2/sqrt3
2/sqrt3 is outside of interval so just use 2/sqrt3 (I think that's considered a critical point)
f(2/sqrt3)= 1.921
f(0) = 5
f(5)=110
I'm not sure what to do with all these numbers. I think since 1.921 is the smallest of the numbers that 2/sqrt 3 is the minimum, but is it the local or absolute.
It looks like 5 is the max, but again local or absolute and can I really include this to be max since it isn't even included in the interval [0,5)
Can someone give me some guidance or show me some steps to solve this.

Functionn has local maximum or minimum where first derivation=0
If second derivate<0 that is local maximum
If second derivate>0 that is local minimum
First derivation=3x^24=0
x=+/ 2/sqroot(3)
sqroot(3)=1.7320508075688772935274463415059
x=+/ 2/1.7320508075688772935274463415059=
1.1547005383792515290182975610039
Second derivate=6x
6*1.1547005383792515290182975610039=
6.9282032302755091741097853660235>0
Function has local maximum for x=1.1547005383792515290182975610039
f(x)max=f(1.1547005383792515290182975610039)=
1.9207985643219959226178731706562
If you want to see graphs of function in google type:
functios graphs online
When you see list of results click on:
rechneronline.de/functiongraphs/
When page be vopen in blue rectangle type:
x^2x^34x+5
In Range xaxis from type 4 to 6
In Range yaxis from type 1 to 9
Then click option Draw 
Sorry:
Second derivate=6x
6*1.1547005383792515290182975610039=
6.9282032302755091741097853660235>0
Function has local minimum for x=1.1547005383792515290182975610039
f(x)min=f(1.1547005383792515290182975610039)=
1.9207985643219959226178731706562 
So the local min and max are both 2/sqrt3. what about the absolite values or do the not exists?
Thanks. 
Bosnian,
Thanks for your help. I just want to make sure I understand. The local min and max are the same f(2/sqrt3) = 1.92
Are there any absolute maximum or minimum?
Thanks again. 
If you try to see graph of that function you can see that function:
x^34x+5
havent absolute maximum or absolute minimum.