y = (x^2 + 1) / x^2
y'
= [x^2 (2x) - (x^2 + 1)(2x)] / x^4
= [2x^3 - 2x^3 - 2x] / x^4
= -2x / x^4
= -2 / x^3
Setting y' to zero, we will see that there is no solution.
Thus, the function does not have a relative maximum or minimum.
Now, we will solve the problem by observation.
y does not exists at x = 0, since x^2 is in the denominator.
lim(x->0) will give you infinity.
lim(x->infinity) will give you zero, and lim(x-> negative infinity) will also give you zero.
Since there are no relative maximum / minimum, this implies that y concaves upwards everywhere except at x = 0.
Hence, y concaves upwards on the intervals (-infinity, 0) and (0, infinity).