Critical Point Intervals- Calculus

Question

I'm having a lot of trouble with the last part of this problem where it asks for four intervals. I have added the correct answers of the other parts and they are underlined.

Find the critical points and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to each critical point. Let 

𝑓(𝑥)= 3/4𝑥^4+15/3𝑥^3+(−48/2)𝑥^2−240𝑥

There are three critical points. If we call them 𝑐1,𝑐2, and 𝑐3, with 

𝑐1< 𝑐2< 𝑐3, then 

𝑐1 =  -5

𝑐2 =  -4

and 𝑐3 = 4 

Is 𝑓 a maximum or minumum at the critical points?

At 𝑐1, 𝑓 is     Local Min        

At 𝑐2, 𝑓 is     Local Max    

At 𝑐3, 𝑓 is     Local Min   

These three critical give us four intervals. ( this is the part that I need help on, I already got the underlined portions right but I need the intervals, which I left an open space for)

The left-most interval is   , and on this interval 𝑓 is    Decreasing    while 𝑓′ is       Negative     .

The next interval (going left to right) is  . On this interval 𝑓 is      Increasing   while 𝑓′ is    Positive      

Next is the interval   . On this interval 𝑓 is    Decreasing     while 𝑓′ is   Negative     

Finally, the right-most interval is   . On this interval 𝑓 is       Increasing   while 𝑓′ is    Positive    

Thank you!

Answer 1

 An interval is just a range of x-values. So, for example, if you were to draw a number line, and then put your critical numbers on the number line, you would see that the number line is now divided into 4 sections. The intervals are going to be the endpoints for each of the four sections. The last thing to keep in mind when writing your answer is whether the endpoints are included or excluded in the intervals. I won't tell you the actual answer, but hopefully this helps you put the pieces together.