Question

For what values of x is `f(x) = -x^4-9x^2-x+5` concave or convex?

Answer 1

Convex on all `x`

Explanation:

We take the second derivative.

`f'(x) = -4x^3 - 18x - 1`

`f''(x) = -12x^2 - 18`

We set to `0`.

`0 = -12x^2 - 18`

`0 = -6(2x^2 + 3)`

`2x^2 + 3 = 0`

`2x^2 = -3`

`x = +- sqrt(-3/2)`

`x = O/`

So there are no inflection points. Since `y = -12x^2 - 18` is negative for all `x`, `f(x)` will be convex on all `x`.

Hopefully this helps!