Question

For what values of x is `f(x)= 2x^3+5x+12` concave or convex?

Answer 1

`f(x)` is concave when `x in ]-oo,0[`
`f(x)` is convex when `x in ]0,+oo[`

Explanation:

We calculate the first and second derivatives

`f(x)=2x^3+5x+12`

`f'(x)=6x^2+5`

`f''(x)=12x`

`f'(x)>0`

`f''(x)=0`, when `x=0`

We draw a chart

`color(white)(aaaa)` `Interval` `color(white)(aaaaaaa)` `]-oo,0[` `color(white)(aaaa)` `]0,+oo[`

`color(white)(aaaa)` `Sign f''(x)` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaaaaa)` `+`

`color(white)(aaaa)` `function` `color(white)(aaaaaaaaaa)` `nnn` `color(white)(aaaaaaaa)` `uuu`

Therefore,

`f(x)` is concave when `x in ]-oo,0[`

`f(x)` is convex when `x in ]0,+oo[`