Question

Is `f(x)=(1-e^(2/x))/(2x-4)` increasing or decreasing at `x=-1`?

Answer 1

Since the derivative is negative, we can conclude that at this point the function is decreasing.

Explanation:

We can use the derivative to give us a slope at a single point (`x=-1`):
`f(x) = (1-e^(2/x))/(2x-4)`
`f'(x)=(df(x))/(dx) = (e^(2/x)(x^2+2x-4)-x^2)/(2(x-2)^2 x^2)`

Assessing the derivative at `x=-1`:
`f'(-1) = -(5 + e^2)/(18 e^2)`

Since the derivative is negative, we can conclude that at this point the function is decreasing.