Please answer as much as possible: If you are asked for a function, enter a func

Question

Please answer as much as possible:

If you are asked for a function, enter a function. If you are asked to find x- or y-values, enter either a number or a list of numbers separated by commas. If there are no solutions, enter None. If you are asked to find an interval or union of intervals, use interval notation. Enter { } if an interval is empty. If you are asked to find a limit, enter either a number, I for infinity, -I for - infinity, or DNE if the limit does not exist. Calculate the first derivative of f . Find the critical numbers of f , where it is increasing and decreasing, and its local extrema. F (x) = Critical numbers x = Increasing on the interval Decreasing on the interval Local maxima x = Local minima x = Find the following left- and right-hand limits at the vertical asymptote x = -2 Find the following left- and right-hand limits at the vertical asymptote x =2- Find the following limits at infinity to determine any horizontal asymptotes. Calculate the second derivative of f. Find where f is concave up, concave down, and has inflection points. f (x) = Concave up on the interval Concave down on the interval Inflection points x = The function f is because for all x in the domain of f, and therefore its graph is symmetric about the ? Answer the following questions about the function f and its graph. The domain of f is the interval The range of f is the interval y-intercept x-intercepts

Explanation / Answer

a)

f'(x)= -48x/(x^2-4)^2

Critical numbers=0

increasing in (-infinity,0)

decreasing in (0,infinity)

maxima at x=0

mimima at no value i.e {}

b)

at x-->-2- we get+infinity

at x-->-2+ we get -infinity

at x-->2- we get -infinity

at x-->2+ we get +infinity

at x-->-infinity we get 6

at x-->infinit we get 6

c)

f''(x)=48(4+3x^2)/(x^2-4)^3

concave up for x>2 or x<-2 i.e (-infinity,-2)U(2,infinity)

concave down for -2<x<2 i.e (-2,2)

point of inflection is none i.e {} because -2 and 2 are not in domain

d)

the function is even as f(x)=f(-x) so its graph is symmetric about y axis

e)

domain is (-infinity,-2) U (2,infinity) i.e (-infinity,infinity)-{-2,2}

y=6x^2/(x^2-4)

So

(6-y)x^2+4y=0

So

x=sqrt(4y/(y-6))

range is y<=0 or y>6 i.e (-infinity,infinity)-(0,6]

y intercept is at x=0

So y intercept is 0

and x intercept is 0

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