Question

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

Answer 1

The function is concave.

Explanation:

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

`f'(x)=-2x+4`

For critical points, `f'(x)=0`

`-2x+4=0`

`x=2`

Let's do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `f'(x)` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `uarr` `color(white)(aa)` `6` `color(white)(aaa)` `darr`

So `f(x)` is concave between `-oo` and `+oo`

graph{-x^2+4x+2 [-16.02, 16.02, -8.01, 8.01]}