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Question: For this problem, consider the function y = f(x) = x^(1/3)(x^2 - 22) on the domain of all real numbers. (a) The value of lim(x→∞) f(x) is _ (If you need to use -∞ or ∞, enter -infinity or infinity.) (b) The value of lim(x→-∞) f(x) is _ (If you need to use -∞ or ∞, enter -infinity or infinity.) (c) There are three x-intercepts; list these in increasing order: s = t = u = (d) The intercepts in part (c) divide the number line into four intervals. From left to right, indicate the sign of the function on each of these intervals by typing "P" (for positive) or "N" (for negative): sign on (-∞,s) = , sign on (s,t) = , sign on (t,u) = , sign on (u,∞) = (e) What is the formula for f '(x) = (f) There are three critical numbers for f(x); list these in increasing order: v = , w = , z = (g) The critical numbers in (f) break the domain into four intervals; on each, determine if the function is increasing or decreasing by typing "I" or "D": sign on (-∞,v) = , sign on (v,w) = , sign on (w,z) = , sign on (z,∞) = (h) There is a local minimum for f(x) occurring at x = (i) There is a local maximum for f(x) occurring at x = (j) lim(x→0) f '(x) = . (If you need to use -∞ or ∞, enter -infinity or infinity; if the limit does not exist, write "DNE".)

          Question: For this problem, consider the function
y = f(x) = x^(1/3)(x^2 - 22)
on the domain of all real numbers.

(a) The value of 
lim(x→∞) f(x)
is _____ (If you need to use -∞ or ∞, enter
-infinity or infinity.) 

(b) The value of 
lim(x→-∞) f(x)
is _____ (If you need to use -∞ or ∞, enter
-infinity or infinity.) 

(c) There are three x-intercepts; list these in
increasing order:
s =  
t =  
u =  

(d) The intercepts in part (c) divide the number line into four
intervals. From left to right, indicate the sign of the function on
each of these intervals by typing "P" (for positive) or "N" (for
negative): 
sign on (-∞,s) =  
 , 
sign on (s,t) =  
 , 
sign on (t,u) =  
 , 
sign on (u,∞) =  

(e) What is the formula for f '(x) = 

(f) There are three critical numbers for f(x); list these in
increasing order: 
v =  
 , 
w =  
 , 
z =  

(g) The critical numbers in (f) break the domain into four
intervals; on each, determine if the function is increasing or
decreasing by typing "I" or "D": 
sign on (-∞,v) =  
 , 
sign on (v,w) =  
 , 
sign on (w,z) =  
 , 
sign on (z,∞) =  

(h) There is a local minimum for f(x) occurring at
x =  

(i) There is a local maximum for f(x) occurring at
x =  

(j) 
lim(x→0) f '(x)
= 
 . (If you need to use -∞ or ∞, enter -infinity
or infinity; if the limit does not exist, write "DNE".)

Added by Magdalena S.

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