Determining decreasing/increasing intervals

Question

True or False:

 

The f(x)=−2ln(x+3.5) is always increasing for all x>−3.5.

 

Enter "true" for true, or "false" for false.

i cant figure this out because the derivative of this function is -2/x+3.5 yet -3.5 cannot be a critical point because it is not in the domain of the original function. How do i figure this problem out? Do i not use the derivative?

Answer 1

For all values x > 3.5, -2/(x + 3.5) is negative. Therefore the function is always decreasing on that restricted domain.

Answer 2

You are correct, there are no critical points as the derivative is nowhere zero.

-2/(x+3.5) < 0 when x+3.5 >0

that happens when x> -3.5

so the function decreases when x > -3.5

Likewise, the derivative is positive when x+3.5<0

which happens when x<-3.5, so the function increases from (-infinity, -3.5)