Question

For what values of x is `f(x)=4/x^2+1` concave or convex?

Answer 1

`f` is convex on the interval `(-oo,0)uu(0,+oo)`.

Explanation:

The determine when a function is concave or convex, analyze the sign, positive or negative, of the function's second derivative:

• When `f''>0`, then `f` is convex.
• When `f''<0`, then `f` is concave.

So, we first must find `f''`.

Note that we can write `f` as

`f(x)=4x^-2+1`

Now, through the power rule, we see that

`f'(x)=-8x^-3`

`f''(x)=24x^-4=24/x^4`

We must now determine when `24/x^4` is positive or negative.

It's necessary to note that `x^4` will always be positive, so `24/x^4` will also always be positive. Recall that the domain of `f` excludes `0`, so we know that

`f` is convex on the interval `(-oo,0)uu(0,+oo)`.