Consider the following function.

Question

f(x)=x^2/x^2-36

(a) Find the critical numbers of f. (Enter your answers as a comma-separated list.)

x =

(b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

increasing
decreasing

(c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.)

relative maximum	(x, y)	=
relative minimum	(x, y)	=

Answer 1

The derivative is f'(x) = (-72x)/(x^2-36)^2

the critical value is x=0 and the critical point is (0,0)

the function increases when x<0 and decreases when x>0

relative max and min is at (0,0)

the asymptotes are at x=6 and x=-6

the function is positive and increase for x<-6

the function is positive and decreasing for x>6

the function is like a parabola between -6 and 6, increases starting at -6

until it hits (0.0) then decreases (parabola like pointing down and belly up at the origin)