Originally Posted by

**skeske1234** I have questions in regards to:

Use the algorithm for curve sketching to sketch:

a) (2x+1)/(x-1)

b) (2x)/(x^2-25)

A) When I am figuring out the intervals of increase and decrease, I am supposed to use the first derivative test..

f'(x)=-3/(x-1)^2

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..

f''(x)=6/(x-1)^3

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.