Question

How do you know where the graph of f(x) is concave up and where it is concave down for `f(x) = x^3 + x`?

Answer 1

The graph of `f` is concave up on intervals on which `f''(x)` is positive and the graph is concave down where `f''(x)` is negative.

Explanation:

So we need to investigate the sign of `f''(x)`.

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

`f'(x) = 3x^2+1`

`f''(x) = 6x`

Clearly, `f''(x) = 6x` is negative for `x <0` and positive for `x>0`.

So the graph of `f` is concave down on `(-oo,0)` and it is concave up on `(0,oo)`