I have questions in regards to:
Use the algorithm for curve sketching to sketch:
A) When I am figuring out the intervals of increase and decrease, I am supposed to use the first derivative test..
and then find its critical numbers
f'(x)=0 when /
f'(x)=undefined when x=1
and then use the critical numbers to find where it is negative or positive, decreasing or increasing.
x<1 = decreasing
x>1 = decreasing
so f is always decreasing.
However, when I find the concavities, I find that the CU and CD intervals are not related to the intervals of decreasing and increasing..
critical numbers f''(x)=0 when x = /
f''(x)=undefined when x=1
Then do the second derivative test to find the concavities
x<0 = negative, concave down
x>0= positive, concave up
so when I say "f is always decreasing" above in my first derivative test, that doesn't match the second derivative results with CU and CD... I have a CU in there, but if concave up is positive, does it mean it has to increase? How can it increase if f is always decreasing.. OR are these completely different things? please clarify this for me.
I am wondering this similar question (can a concave up can occur when f is ALWAYS decreasing? and can a concave down occur when f is ALWAYS increasing?) with regards to part B as well.
Thanks for your help.