Derivatives/critical values and asymptotes

Question

Consider the function f(x)=x2+3x+2/(x−4)

1. List any critical values.   
2. Identify intervals of increase. 
3. Identify intervals of decrease. 
4. Find the second derivaitve. f''(x)=f′′(x)=  
5. Based on parts b through e, f(x)f(x) has a maximum of y=y= when x=x=
6. Based on parts b through e, f(x)f(x) has a minimum of y=y= when x=x=
7. Use the second derivative to identify intervals where f(x)f(x) is concave up. 
8. Use the second derivative to identify intervals where f(x)f(x) is concave down. 
9. Use the second derivative to find any inflection points. 
10. State any vertical asymptotes.   
11. State any slant asymptotes. 

Answer 1

go to an online graphing calculator, it may help find several of the answers you want as well as be a check on what you get algebraically with derivatives

Desmos shows an always concave downward graph, with x intercepts = -2 and -2

local and global max at x=-3/2

increasing nowhere

decreasing everywhere

max y = (-3/2, 1/22), global and local

no min y

you can read all that off the graph

if you get something different from calculus and algebra calculations, either recheck your arithmetic or check with another graphing calculator. At least 2 graphing calculators show the same solutions, same graph

f(x) = (x^2+3x+2)/(x-4) has a vertical asymptote when the denominator =0: x=4

horizontal asymptote not asked

max y is at (-3/2, 1/22)

there is no minimum y as it approaches negative infinity as x goes to + or - infinity

f"(x) = 4+/- sqr30

f"(x)= when the concavity changes. It never changes. there is no inflection point

f"(x) = 0 for all x

Answer 2

desmos.com/calculator/7k9lsjcxza