Question

Is `f(x)=1/x-1/x^3+1/x^5` increasing or decreasing at `x=1`?

Answer 1

I think is decreasing.

Explanation:

We can first evaluate the derivative of your function `f'(x)` which gives the inclination of your function; then we evaluate the derivative at `x=1` to see if:
`f'(1)>0` the function is increasing;
`f'(1)<0` the function is decreasing;
`f'(1)=0` the function has a minimum/maximum.

So:
`f'(x)=-1/x^2+3/x^4-5/x^6`

Evaluate it at `x=1`
`f'(1)=-1+3-5=-3<0`
the function is decreasing.

Graphically:
graph{(1/x)-(1/x^3)+(1/x^5) [-0.758, 4.11, -1.14, 1.292]}