Which of the following could represent a function having the given properties?

Question

1)increasing slope for x<4
2) f'(x)>0 for all x(x cannot =4)
3)asymptote at x=4.

Again I know we cannot graph but how can I figure this out?

Answer 1

Asymptote at x=4 indicates a factor of (x-4) in the denominator.

Let's try f(x)=1/(x-4)

f'(x)=-1/(x-4)²
so f'(x)<0 for all x except 4.

Let's try f(x)=-1/(x-4)
f'(x)=1/(x-4)² >0 for all x.

Check f"(x)=2/(x-4)³
f"(x)>0 ∀x<4.

So all conditions are satisfied.