Question

How do you determine the intervals where `f(x)=x^(2/3)+3` is concave up or down?

Answer 1

Investigate the sign of the second derivative.

Explanation:

`f(x) = x^(2/3)+3`

The domain of `f` is all real numbers.

`f'(x) = 2/3x^(-1/3)`

`f''(x) = -2/9 x^(-4/3) = -2/(9root(3)x^4)`

`f''` is never `0` and is undefined at `x=0`.

So the only place where `f''` might change signs is `x=0`

On `(-oo,0)`, we see that `f''` is negative, so `f` is concave down.

On `(0,oo)`, we see again that `f''` is negative, so `f` is concave down.

Because the concavity does not change, there is no inflection point.

By the way, here is the graph of `f(x)`.

graph{y=x^(2/3)+3 [-12.62, 12.69, -2.98, 9.68]}