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

1 Answer
Dec 7, 2015

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]}