find intervals on a graph

Question

Find all intervals on which the graph of y= ((x^2+1)/x^2) is concave upward?

Answer 1

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).