Determine the open intervals on which the graph of f(x) =X2 +1/ X2 -4 is concave upward or concave downward.

Question

Determine the open intervals on which the graph of f(x) =X² +1/ X² -4 is concave upward or concave downward.

Answer 1

f(x) = (x^2+1)/(x^2-4)

f'(x) = ((x^2-4)(2x) - (x^2+1)(2x))/(x^2-4)^2

= (2x)(-5)/(x^2-4)^2

= -10x/(x^4 -8x^2 +16)

f"(x) =-10x^4 +40x^3+80x^2-160x-160/d(x-2)^4 =0

x=+/-2

where the concavity changes

the three intervals are

x<-2

-2 <x< 2

x> 2

or in interval notation

(-infinity,-2),(-2, 2), (2,infinity)

1st interval concave up,

2nd interval concave down,

3rd interval concave up

find second derivative, set=0, solve for x

or use a graphing calculator

or just plot a few points and sketch a graph

asymptotes are x=+/-2 and y=1