Question

for f(x)=(x^4+1)/(x^2) find any maximum, minimum, points of inflection, intervals where the function may be increasing or decreasing, concave up, or concave down. please help and show all steps!

Answer 1

This problem requires quite a bit of work. It's not too difficult, but the process is rather lengthy, so I will only outline it briefly since you are not present:

max/min occur when f'(x)=0, so solve that
inflection points occur when f''(x)=0, so solve that.

having found those roots will break up your function into intervals between those points, which can be used to check to see if the graph pf f(x) is increasing, decreasing, concave up, or concave down.
check a point in each interval by plugging in some value x=c (c is just some constant in that interval) in that interval.

The results of that point-testing will determine the shape and behavior of f(x) as follows:
if f'(c)<0 then f(x) is decreasing on that interval
if f'(c)>0 then f(x) is increasing on that interval
if f''(c)<0 then f(x) is concave down on that interval
if f''(c)>0 then f(x) is concave up on that interval