An even function f has a vertical asymptote at x = 3 and a maximum at x = 0. Which of the following could be f?

Question

a. f(x)=(x)/(x^2-9)
b. f(x)=(x)/(x-3)
c. f(x)=(x^2)/(x^2+9)
d. f(x)=(x^2)/(x^4-81)

Answer 1

First determine which of the cases has an asymptote at x=3:
Only a, b and d have a factor of (x+3) in the denominator. So c is out.

Since it has a maximum at x=0, chances are that odd functions are out. That takes away a and b.

For the remaining (d), diffentiate with respect to x and equate to zero to see if there is a maximum at zero.

Alternatively, since we know
Lim f(0)=0
x->0
we can try evaluating f(x) at f(0-) and f(0+).
It is easy to see that, since (d) is an even function, even without a calculator, both f(0-) and f(0+) are <0, thus f(0) is a maximum.